Worksheets for GCSE Mathematics. Quadratics. mrmathematics.com Maths Resources for Teachers. Algebra


 Donald Lewis
 2 years ago
 Views:
Transcription
1 Worksheets for GCSE Mathematics Quadratics mrmathematics.com Maths Resources for Teachers Algebra
2 Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation Solving ax + bx + c by factorisation Completing the Square Quadratic Formula Quadratic and Linear Simultaneous Equations Sketching Quadratics Page 30 Page 40 Page 50 Page 60 Page 70 Page 80 Solutions Solving x + bx + c by factorisation Solving ax + bx + c by factorisation Completing the Square Quadratic Formula Quadratic and Linear Simultaneous Equations Sketching Quadratics Page 9 Page 10 Page 11 Page 1 Page 13 Page 14
3 Q1. Solve these quadratic equations. Solving Quadratics by Factorising a) xx(xx + 3) = 0 b) xx(xx 5) = 0 c) xx(xx 6) = 0 d) xx + xx = 0 e) xx + 1xx = 0 f) xx + 7xx = 0 Q. Solve these equations by factorising the quadratic. a) kk + 7kk + 10 = 0 b) aa + 8aa + 16 = 0 c) xx + 4xx + 3 = 0 d) jj + 13jj + = 0 e) vv + vv 4 = 0 f) mm 9mm + 0 = 0 g) bb + bb 30 = 0 h) ff 8ff + 7 = 0 i) qq 10qq + 5 = 0 j) cc 6cc 16 = 0 k) ww 5ww 84 = 0 l) tt 18tt + 80 = 0 m) gg 5 = 0 n) h 49 = 0 o) xx 64 = 0 Q3. Use the method of factorisation to solve these quadratics. a) xx + 4xx = 3 b) ff = 16 c) nn nn = LE d) yy = 9yy 0 e) vv + 1 = vv f) h + 6 = 5h g) xx + 1 = 7xx h) 14dd 49 = dd i) xx aa = 0 Q4. a) The length of a pool is 8 m more than its width. The area is 65 m. Calculate the dimensions of the pool. b) The base of a triangle is 7 cm greater than its perpendicular height. The area is 60 cm. Calculate the perpendicular height. c) The sum of the squares of two consecutive positive integers is 65. i) Show that xx + xx 13 = 0. ii) Calculate the two integers. d) The difference between a number and its square is 7. i) Show that xx xx 7 = 0 ii) Calculate possible values of x. 3
4 Q1. Match the solutions for xx to each equation. Solving aaxx + bbbb + cc = 0 by Factorising a) (3xx + 1)(xx 5) = 0 i) xx = 1, xx = 3 5 b) (5xx 1)(xx + 3) = 0 ii) xx = 1 3, xx = 5 c) (3xx 1)(xx + 5) = 0 iii) xx = 3 5, xx = 3 d) (5xx + 3)(3 xx) = 0 iv) xx = 1, xx = 5 3 Q. Use factorisation to solve the following quadratic equations. a) xx + 9xx + 4 = 0 b) xx + 13xx + 15 = 0 c) 3xx + 10xx + 8 = 0 d) 3xx + xx = 0 e) 3xx 13xx + 14 = 0 f) 5xx + 7xx 6 = 0 g) xx xx + 3 = 0 h) 3 5xx xx = 0 Q3. Use factorisation to solve the following quadratic equations. LE a) 6xx + 7xx + = 0 b) 4xx 1 = 0 c) 4xx + 1xx + 5 = 0 d) 6xx 15xx 9 = 0 e) 8xx + 6xx + 1 = 0 f) 10xx + 7xx 6 = 0 Q4. Use algebra to find the coordinates of the roots for each equation. a) b) Q5. Solve the following quadratic equations. 4 a) 4xx + 3 = 4xx b) 9xx + 5 = 30xx c) 6xx = 7xx + 3 d) 10xx = 3xx 1
5 Completing the Square Q1. Use the method of completing the square to solve these equations. Leave your answer in its most exact form. a) xx 6xx + 9 = 0 b) xx + 4xx 1 = 0 c) xx + xx 1 = 0 d) xx 4xx 7 = 0 e) xx 10xx 5 = 0 f) xx + 9xx + 1 = 0 g) xx xx 1 = 0 h) xx + 3xx = 0 i) xx 5xx 10 = 0 Q. Use the method of completing the square to solve these equations. Leave your answer in its most exact form. a) xx + xx 1 = 0 b) 4xx xx 8 = 0 c) xx + 6xx 13 = 0 d) 3xx 10xx = 0 e) 3xx 8xx = 0 f) 5xx 3xx 1 = 0 Q3. Use the method of completing the square to solve the following quadratic identities. a) xx + 4xx + 9 (xx + gg) + ff b) xx + 1xx + 3 (xx + bb) + cc LE c) xx 10xx + 34 (xx + jj) + kk d) 3xx + 1xx + 5 rr(xx + ss) + tt e) 3xx + 6xx 1 aa(xx + bb) + cc f) 4xx 16xx + 3 ee(xx + ff) + gg Q4. Use the method of completing the square to determine coordinates of the turning point for each parabola. a) yy = xx + 3xx 5 b) yy = 5xx 3 xx 5
6 Using the Quadratic Formula Q1. Use the Quadratic Formula to solve these equations. Give your answers in their most exact form. a) xx + 8xx + 16 = 0 b) yy 5yy + 4 = 0 c) h 10h + 1 = 0 d) aa + 3aa + 1 = 0 e) ww 3ww = 0 f) 3bb + 10bb = 0 g) 7rr + 9rr + 1 = 0 h) 5qq + qq = 0 i) 4tt + 7tt 6 = 0 Q. Solve the following quadratic equations. Write your answers correct to two decimal places. a) xx + 10xx 5 = 0 b) cc 9cc + 5 = 0 c) kk + 4kk 3 = 0 d) 5ee 6ee = e) 4dd 3 = 9dd f) jj = 8jj 1 g) 16vv = 4vv + 5 h) h = 5h + 3 i) 7aa + = 8aa Q3. Simplify and solve the following equations. Write your answers correct to two decimal places. a) yy(yy 3) = 5 b) (ww 5) = 8 c) xx(3xx 1) = 4xx + d) 1 gg + gg = 5 e) + 3rr = 7 f) = 5(mm 1) rr mm Q4. a) Billy is 3 years younger than his sister. The product of their ages is 38. Find each of their ages. b) The width of a rectangle is three units longer than its length. The area of the rectangle is 30 units. Calculate the perimeter of the rectangle to three significant figures. c) LE The area of the trapezium is 80 units. a) Show that the area of the trapezium is: xx + 3xx 38 = 0 b) Calculate the longest length of the trapezium. 6 Quadratic Formula xx + bbbb + cc = 0 xx = bb ± bb 4aaaa aa
7 Q1. Use the method of substitution to find the intersection of these graphs. Quadratic and Linear Simultaneous Equations yy = xx + 5 yy = xx xx Q. Use the method of substitution to find the intersection of these graphs. LE yy = xx + 1 xx + yy = 16 Q3. Use the method of substitution to solve each pair of simultaneous equations. a) yy = xx + 5 yy = xx xx 8 d) yy = xx + 1 xx + yy = 9 g) xxxx + xx = 5 7 xx + yy = 1 b) yy = 5xx 4 yy = xx + 4xx 16 e) yy = 4xx 3 xx + yy = 4 h) 1 xx = yy xx + yy = 7 c) yy = 3 xx yy = xx + xx 6 f) yy = xx xx + 7yy = i) yy = 3xx xx + yy = 3
8 Q1. Match the equation to each graph. Sketching Quadratics a) b) c) d) LE i) yy = xx 4 ii) yy = (xx + ) iii) yy = 4 xx iv) yy = (xx ) Q. Sketch the graphs of the following functions. 8 State the roots, yintercept and turning point for each graph. a) yy = xx 9 b) yy = xx 3xx + 10 c) yy = xx + xx 15 d) yy = xx 3xx + e) yy = 1 xx xx f) yy = xx 3xx 4 g) yy = 7 + 4xx 3xx
9 Solutions Q1. Solving Quadratics by Factorising a) xx = 0, xx = 3 b) xx = 0, xx = 5 c) xx = 0, xx = 6 d) xx = 0, xx = 1 e) xx = 0, xx = 1 f) xx = 0, xx = 7 Q. a)kk = 5, kk = b) aa = 4 c) xx = 3, xx = 1 d) jj = 11, jj = e) vv = 6, vv = 4 f) mm = 4, mm = 5 g) bb = 6, bb = 5 h) ff = 1, ff = 7 i) qq = 5 j) cc =, cc = 8 k) ww = 7, ww = 1 l) tt = 8, tt = 10 m) gg = 5, gg = 5 n) h = 7, h = 7 o) xx = 8, xx = 8 Q3. LE a) xx = 3, xx = 1 b) ff = 4, ff = 4 c) nn = 1, nn = d) yy 4, yy = 5 e) vv = 1 f) h = 3, h = g) xx = 3, xx = 4 h) dd = 7 i) xx = aa, xx = aa Q4. a) Length = 5 cm, Width = 13 cm b) Height = 8 cm c) ii) x = 11, x = 1 d) ii) x = 8, x = 9 9
10 Solutions Solving aaxx + bbbb + cc = 0 by Factorising Q1. a > ii b > i c > iv d > iii Q. a) xx = 4, xx = 0.5 b) xx = 5, xx = 1.5 Q3. c) xx =, xx = 4 3 e) xx =, xx = 7 3 d) xx = 1, xx = 3 f) xx =, xx = 3 5 g) xx = 1.5, xx = 1 h) xx = 3, xx = 0.5 a) xx = 3, xx = 1 c) xx = 5, xx = 1 e) xx = 1, xx = 1 4 b) xx = 1, xx = 1 d) xx = 1, xx = 3 f) xx = 6 5, xx = 1 Q4. a) 3, 0, (4, 0) b) 3, 0, ( 3, 0) LE Q5. a) xx =, xx = 4 b) xx = c) xx = 1 3, xx = 3 d) xx = 4 5, xx = 3
11 Solutions Q1. Completing the Square a) xx = 3 b) xx = 7 & 3 c) xx = 1 ± d) xx = ± 11 e) xx = 5 ± 30 g) xx = 1± 5 Q. a) xx = 3 1 d) xx = 5± 31 Q3. 3 h) xx = 3± 17 b) xx = 1± 19 8 e) xx = 4± LE 3 f) xx = 9± 33 i) xx = 5± 65 c) xx = 3± 35 f) xx = 3± 9 a) xx + 4xx + 9 (xx + ) + 55 b) xx + 1xx + 3 (xx + 66) 44 c) xx 10xx + 34 (xx 55) + 99 d) 3xx + 1xx (xx + ) 77 e) 3xx + 6xx 1 33(xx + 11) 44 f) 4xx 16xx (xx ) + 77 Q4. a) Minimum = (1.5,7.5) b) Maximum = (1.5, 0.15) 11 10
12 Q1. Using the Quadratic Formula a) xx = 4 & 4 b) yy = 1 & 4 c) h = 3 & 7 d) aa = 3± 5 g) rr = 9± 53 Q. 14 e) ww = 3± 17 f) bb = 5± 31 h) qq = 1± i) tt = 7± 145 a) xx = & 0.48 b) cc = 0.65 & 3.85 c) kk =.58 & 0.58 d) ee = 1.47 & 0.7 e) dd = 0.9 &.54 f) jj = 6 & g) vv = 0.34 & 3.66 h) h = 3 & 0.5 i) aa = 0.77 & 0.37 Q3. a) yy = 4.19 & 1.19 b) ww = 7.83 &.17 c) xx = 0.33 & d) gg =.8 & 0. e) rr = & 0.33 f) mm = 0.31 & 1.31 Q4. LE a) Billy = 14 Sister = 17 b) Length = 4.17 units, Width = 7.17 units. Perimeter =.7 units. c) xx = 4.84 uuuuuuuuuu. Longest Length = 0.53 units 1 3 8
13 Solutions Q1. xx = 1.45, yy = 3.55 xx = 3.45, yy = 8.45 Q. xx = 3.8,.8 xx =.8, yy = 3.8 Q3. a) xx =.41, yy =.59 xx = 5.41, yy = Quadratic and Linear Simultaneous Equations b) xx = 3, yy = 19 xx = 4, yy = 16 LE d) xx =.56, yy = 1.56 xx = 1.56, yy =.56 g) xx =, yy = 1.5, xx = 1.66, yy = e) xx = 0.5, yy = 1.98 xx = 1.16, yy = 1.63 h) xx = 0.77, yy =.53 xx = 1.57, yy =.13 c) xx = 5.61, yy = 14.1 xx = 1.61, yy = 0.1 f) xx = 14.14, yy = 8.8 xx = 0.14, yy = 0.8 i) xx = 0.5, yy = 1.57 xx = 0.5, yy = 1.57
14 Solutions Sketching Quadratics Q1. a = i b = iv c = iii d = ii Q a) yy = xx 9 b) yy = xx 3xx + 10 c) yy = xx + xx 15 d) yy = xx 3xx + LE e) yy = 1 xx xx f) yy = xx 3xx 4 g) yy = 7 + 4xx 3xx 14
15 15 Sketching Quadratics LE
Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 19. Algebra
Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 19 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets
More information10.1 Three Dimensional Space
Math 172 Chapter 10A notes Page 1 of 12 10.1 Three Dimensional Space 2D space 0 xx.. xx, 0 yy yy, PP(xx, yy) [Fig. 1] Point PP represented by (xx, yy), an ordered pair of real nos. Set of all ordered
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationLesson 15: Rearranging Formulas
Exploratory Challenge Rearranging Familiar Formulas 1. The area AA of a rectangle is 25 in 2. The formula for area is AA = llll. A. If the width ll is 10 inches, what is the length ll? AA = 25 in 2 ll
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationWorksheets for GCSE Mathematics. Solving Equations. Mr Black's Maths Resources for Teachers GCSE 19. Algebra
Worksheets for GCSE Mathematics Solving Equations Mr Black's Maths Resources for Teachers GCSE 19 Algebra Equations Worksheets Contents Differentiated Independent Learning Worksheets Solving Equations
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationQuadratic Equations and Functions
50 Quadratic Equations and Functions In this chapter, we discuss various ways of solving quadratic equations, aaxx 2 + bbbb + cc 0, including equations quadratic in form, such as xx 2 + xx 1 20 0, and
More informationIntegrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster
Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of
More informationSection 5: Quadratic Equations and Functions Part 1
Section 5: Quadratic Equations and Functions Part 1 Topic 1: RealWorld Examples of Quadratic Functions... 121 Topic 2: Factoring Quadratic Expressions... 125 Topic 3: Solving Quadratic Equations by Factoring...
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationMath 3 Unit 4: Rational Functions
Math Unit : Rational Functions Unit Title Standards. Equivalent Rational Expressions A.APR.6. Multiplying and Dividing Rational Expressions A.APR.7. Adding and Subtracting Rational Expressions A.APR.7.
More informationSystems of Linear Equations
Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
More informationLesson 13: More Factoring Strategies for Quadratic Equations & Expressions
: More Factoring Strategies for Quadratic Equations & Expressions Opening Exploration Looking for Signs In the last lesson, we focused on quadratic equations where all the terms were positive. Juan s examples
More informationSecondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet
Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and
More informationCore Mathematics 1 Quadratics
Regent College Maths Department Core Mathematics 1 Quadratics Quadratics September 011 C1 Note Quadratic functions and their graphs. The graph of y ax bx c. (i) a 0 (ii) a 0 The turning point can be determined
More informationSection A Plotting Straight Line Graphs Grade D / C
Name: Teacher Assessment Section A Plotting Straight Line Graphs Grade D / C 1. (a) Complete the table of values for = 3x + x 0 1 3 5 10 16 19 (b) On the grid draw the graph of = 3x + for values of x from
More informationSAMPLE COURSE OUTLINE MATHEMATICS METHODS ATAR YEAR 11
SAMPLE COURSE OUTLINE MATHEMATICS METHODS ATAR YEAR 11 Copyright School Curriculum and Standards Authority, 2017 This document apart from any third party copyright material contained in it may be freely
More informationTransition to College Math and Statistics
Transition to College Math and Statistics Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear College Algebra Students, This assignment
More informationSection 2: Equations and Inequalities
Topic 1: Equations: True or False?... 29 Topic 2: Identifying Properties When Solving Equations... 31 Topic 3: Solving Equations... 34 Topic 4: Solving Equations Using the Zero Product Property... 36 Topic
More informationLesson 24: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise 1. Examine the two equation below and discuss what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. cc 14 = 5cc. Solve each equation with the
More informationF.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 ZeroProduct Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or firstdegree, equations. Solving higher
More informationPreparing for the HNC Electrical Maths Components. online learning. Page 1 of 15
online learning Preparing for the HNC Electrical Maths Components Page 1 of 15 Contents INTRODUCTION... 3 1 Algebraic Methods... 4 1.1 Indices and Logarithms... 4 1.1.1 Indices... 4 1.1.2 Logarithms...
More informationUnit Calendar. Date Sect. Topic Homework HW OnTime Apr , 2, 3 Quadratic Equations & Page 638: 311 Page 647: 329, odd
Name/Period: Unit Calendar Date Sect. Topic Homework HW OnTime Apr. 4 10.1, 2, 3 Quadratic Equations & Page 638: 311 Graphs Page 647: 329, odd Apr. 6 9.4 10.4 Solving Quadratic Equations by Factoring
More informationdue date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)
Honors PreCalculus Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear Honors PreCalculus Students, This assignment is designed
More informationP.3 Division of Polynomials
00 section P3 P.3 Division of Polynomials In this section we will discuss dividing polynomials. The result of division of polynomials is not always a polynomial. For example, xx + 1 divided by xx becomes
More informationPLC Papers Created For:
PLC Papers Created For: Year 11 Topic Practice Paper: Factorising Quadratics Factorising difficult quadratic expressions 1 Grade 7 Objective: Factorise a quadratic expression of the form ax 2 + bx + c
More informationLesson 25: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise Over the years, many students and teachers have thought of ways to help us all remember the quadratic formula. Below is the YouTube link to a video created by two teachers
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationNational 5 Mathematics. Practice Paper E. Worked Solutions
National 5 Mathematics Practice Paper E Worked Solutions Paper One: NonCalculator Copyright www.national5maths.co.uk 2015. All rights reserved. SQA Past Papers & Specimen Papers Working through SQA Past
More informationF.1 Greatest Common Factor and Factoring by Grouping
1 Factoring Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers.
More informationSection 6 Quadratics Part 1
Section 6 Quadratics Part 1 The following Mathematics Florida Standards will be covered in this section: MAFS.912.ASSE.1.2 Use the structure of an expression to identify ways to rewrite it. For example,
More informationMathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.
Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...
More informationPLC Papers Created For:
PLC Papers Created For: Quadratics intervention Deduce quadratic roots algebraically 1 Grade 6 Objective: Deduce roots algebraically. Question 1. Factorise and solve the equation x 2 8x + 15 = 0 Question
More informationAlgebra I Chapter 4 Curriculum and IXL
Chapter 4 Curriculum and IXL C4L1 Functions and NonFunctions Represent relations as mappings, sets of points, and graphs: WS Determine whether a relation is a function or not: WS C4L2 Linear and NonLinear
More informationBHASVIC MαTHS. Skills 1
PART A: Integrate the following functions with respect to x: (a) cos 2 2xx (b) tan 2 xx (c) (d) 2 PART B: Find: (a) (b) (c) xx 1 2 cosec 2 2xx 2 cot 2xx (d) 2cccccccccc2 2xx 2 ccccccccc 5 dddd Skills 1
More informationMath, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD
Math, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD Outline These preliminaries serve to signal to students what tools they need to know to succeed in ECON 360 and refresh their
More informationKing Fahd University of Petroleum and Minerals PrepYear Math Program Math (001)  Term 181 Recitation (1.1)
Recitation (1.1) Question 1: Find a point on the yaxis that is equidistant from the points (5, 5) and (1, 1) Question 2: Find the distance between the points P(2 x, 7 x) and Q( 2 x, 4 x) where x 0. Question
More informationLesson 23: The Defining Equation of a Line
Classwork Exploratory Challenge/Exercises 1 3 1. Sketch the graph of the equation 9xx +3yy = 18 using intercepts. Then, answer parts (a) (f) that follow. a. Sketch the graph of the equation yy = 3xx +6
More information1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
More informationFinding the Equation of a Graph. I can give the equation of a curve given just the roots.
National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x  8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today
More informationSpecialist Mathematics 2019 v1.2
Examination (15%) This sample has been compiled by the QCAA to assist and support teachers in planning and developing assessment instruments for individual school settings. The examination must ensure
More informationAlgebra 1 (cp) Midterm Review Name: Date: Period:
Algebra 1 (cp) Midterm Review Name: Date: Period: Chapter 1 1. Evaluate the variable expression when j 4. j 44 [1] 2. Evaluate the variable expression when j 4. 24 j [2] 3. Find the perimeter of the rectangle.
More informationA. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?
Hart Interactive Honors Algebra 1 Lesson 6 M4+ Opening Exercises 1. Watch the YouTube video Imaginary Numbers Are Real [Part1: Introduction] by Welch Labs (https://www.youtube.com/watch?v=t647cgsuovu).
More informationLesson 23: Deriving the Quadratic Formula
: Deriving the Quadratic Formula Opening Exercise 1. Solve for xx. xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 Discussion 2. Which of these problems makes more sense to solve by completing the square? Which makes
More informationTerms of Use. Copyright Embark on the Journey
Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means  electronic, mechanical, photocopies, recording, or otherwise
More informationG.6 Function Notation and Evaluating Functions
G.6 Function Notation and Evaluating Functions ff ff() A function is a correspondence that assigns a single value of the range to each value of the domain. Thus, a function can be seen as an inputoutput
More informationVariations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated
More information6.4. The Quadratic Formula. LEARN ABOUT the Math. Selecting a strategy to solve a quadratic equation. 2x 2 + 4x  10 = 0
6.4 The Quadratic Formula YOU WILL NEED graphing calculator GOAL Understand the development of the quadratic formula, and use the quadratic formula to solve quadratic equations. LEARN ABOUT the Math Devlin
More informationChapter 22 : Electric potential
Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts
More informationMathematical Methods 2019 v1.2
Examination This sample has been compiled by the QCAA to model one possible approach to allocating marks in an examination. It matches the examination mark allocations as specified in the syllabus (~ 60%
More information1 Nama:... Kelas :... MAKTAB SABAH, KOTA KINABALU PEPERIKSAAN PERTENGAHAN TAHUN 2009 MATEMATIK TAMBAHAN TINGKATAN 4 Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1. This question
More informationTSIA MATH TEST PREP. Math and Science, ASC 1
TSIA MATH TEST PREP Math and Science, ASC 1 Texas Success Initiative: Mathematics The TSI Assessment is a program designed to help Lone Star College determine if you are ready for collegelevel coursework
More information7.3 The Jacobi and GaussSeidel Iterative Methods
7.3 The Jacobi and GaussSeidel Iterative Methods 1 The Jacobi Method Two assumptions made on Jacobi Method: 1.The system given by aa 11 xx 1 + aa 12 xx 2 + aa 1nn xx nn = bb 1 aa 21 xx 1 + aa 22 xx 2
More informationTest 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.110.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More informationMonday Tuesday Wednesday Thursday
Answer Key  Weekly Math Homework Q3:1 Write the equation of a line in slope intercept form that has a slope of 3 and has a y 5 intercept of ½. yy = 3 5 xx + 1 2 Write the exponential function that matches
More informationTECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMNFOOTING PROPERTIES
COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMNFOOTING PROPERTIES Introduction This technical note
More informationTest of Mathematics for University Admission. Specification for October 2018
Test of Mathematics for University Admission Specification for October 2018 Structure of the Test The test will consist of two 75 minute papers, taken one after the other. Each paper will consist of 20
More informationP.4 Composition of Functions and Graphs of Basic Polynomial Functions
06 section P4 P.4 Composition of Functions and Graphs of Basic Polynomial Functions input ff gg gg ff In the last three sections, we have created new functions with the use of function operations such
More informationMathematics Paper 2 Grade 12 Preliminary Examination 2017
Mathematics Paper 2 Grade 12 Preliminary Examination 2017 DURATION: 180 min EXAMINER: R. Obermeyer MARKS: 150 MODERATOR: A. Janisch Date: 15 September 2017 External Moderator: I. Atteridge INSTRUCTIONS:
More informationSection 3: Introduction to Functions
Topic 1: Input and Output Values... 55 Topic 2: Representing, Naming, and Evaluating Functions... 58 Topic 3: Adding and Subtracting Functions... 60 Topic 4: Multiplying Functions... 62 Topic 5: Closure
More informationSpring 2018 Math Week Week 1 Task List
Spring 2018 Math 143  Week 1 25 Week 1 Task List This week we will cover Sections 1.1 1.4 in your ebook. Work through each of the following tasks, carefully filling in the following pages in your notebook.
More informationLesson 7: Algebraic Expressions The Commutative and Associative Properties
Algebraic Expressions The Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 + 4: Ben looks at this picture from the opposite side of the
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationExam Programme VWO Mathematics A
Exam Programme VWO Mathematics A The exam. The exam programme recognizes the following domains: Domain A Domain B Domain C Domain D Domain E Mathematical skills Algebra and systematic counting Relationships
More informationFurther Mathematics Summer work booklet
Further Mathematics Summer work booklet Further Mathematics tasks 1 Skills You Should Have Below is the list of the skills you should be confident with before starting the ALevel Further Maths course:
More informationF.3 Special Factoring and a General Strategy of Factoring
F.3 Special Factoring and a General Strategy of Factoring Difference of Squares section F4 233 Recall that in Section P2, we considered formulas that provide a shortcut for finding special products, such
More informationNorth Carolina MATH I (Traditional) Pacing Guide
North Carolina MATH I (Traditional) 20182019 Pacing Guide Note: The eight Standards for Mathematical Practice describe the varieties of expertise that mathematics educators should seek to develop in their
More informationSpecialist Mathematics 2019 v1.2
Examination This sample has been compiled by the QCAA to assist and support teachers in planning and developing assessment instruments for individual school settings. Schools develop internal assessments
More informationDepartment of Mathematics, University of WisconsinMadison Math 114 Worksheet Sections 3.1, 3.3, and 3.5
Department of Mathematics, University of WisconsinMadison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the yintercept. f(x) = 5x + is of the form y = mx
More informationIn a quadratic expression the highest power term is a square. E.g. x x 2 2x 5x 2 + x  3
A. Quadratic expressions B. The difference of two squares In a quadratic expression the highest power term is a square. E.g. x + 3x x 5x + x  3 If a quadratic expression has no x term and both terms are
More informationPolynomials and Polynomial Functions
1 Polynomials and Polynomial Functions One of the simplest types of algebraic expressions are polynomials. They are formed only by addition and multiplication of variables and constants. Since both addition
More informationF.1 Greatest Common Factor and Factoring by Grouping
section F1 214 is the reverse process of multiplication. polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example,
More informationNumber Representations
Computer Arithmetic Algorithms Prof. Dae Hyun Kim School of Electrical Engineering and Computer Science Washington State University Number Representations Information Textbook Israel Koren, Computer Arithmetic
More informationA Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:
A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,
More informationMathematical Methods 2019 v1.2
Problemsolving and modelling task (20%) This sample has been compiled by the QCAA to assist and support teachers to match evidence in student responses to the characteristics described in the instrumentspecific
More information= The algebraic expression is 3x 2 x The algebraic expression is x 2 + x. 3. The algebraic expression is x 2 2x.
Chapter 7 Maintaining Mathematical Proficiency (p. 335) 1. 3x 7 x = 3x x 7 = (3 )x 7 = 5x 7. 4r 6 9r 1 = 4r 9r 6 1 = (4 9)r 6 1 = 5r 5 3. 5t 3 t 4 8t = 5t t 8t 3 4 = ( 5 1 8)t 3 4 = ()t ( 1) = t 1 4. 3(s
More informationP.2 Multiplication of Polynomials
1 P.2 Multiplication of Polynomials aa + bb aa + bb As shown in the previous section, addition and subtraction of polynomials results in another polynomial. This means that the set of polynomials is closed
More informationNumerical and Algebraic Fractions
Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More information27 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)
Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,
More informationA A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good.
Victorian Handwriting Sheet Aa A A A A A A A A A A A A Aa Aa Aa Aa Aa Aa Aa a a a a a a a a a a a a a a a Apples taste amazingly good. Apples taste amazingly good. Now make up a sentence of your own using
More informationMrs. Charnley, J303 Mrs. Zayaitz Ruhf, J305
Mrs. Charnley, J303 charnleyc@eastonsd.org Mrs. Zayaitz Ruhf, J305 zayaitzruhfm@eastonsd.org Part I. Online Quiz/Review, DUE June 30, 016 This is a quick review of some material from PreCalculus and other
More informationDefinition: A sequence is a function from a subset of the integers (usually either the set
Math 3336 Section 2.4 Sequences and Summations Sequences Geometric Progression Arithmetic Progression Recurrence Relation Fibonacci Sequence Summations Definition: A sequence is a function from a subset
More informationA101 ASSESSMENT Quadratics, Discriminant, Inequalities 1
Do the questions as a test circle questions you cannot answer Red (1) Solve a) 7x = x 230 b) 4x 229x + 7 = 0 (2) Solve the equation x 2 6x 2 = 0, giving your answers in simplified surd form [3] (3) a)
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationMath 3336 Section 4.3 Primes and Greatest Common Divisors. Prime Numbers and their Properties Conjectures and Open Problems About Primes
Math 3336 Section 4.3 Primes and Greatest Common Divisors Prime Numbers and their Properties Conjectures and Open Problems About Primes Greatest Common Divisors and Least Common Multiples The Euclidian
More informationEureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities
Eureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities These 2 lessons can easily be taught in 2 class periods. If you like these lessons, please consider using other Eureka lessons as well.
More informationChapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)
Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5
More informationLogarithmic Functions
Name Student ID Number Group Name Group Members Logarithmic Functions 1. Solve the equations below. xx = xx = 5. Were you able solve both equations above? If so, was one of the equations easier to solve
More informationAdditional Functions, Conic Sections, and Nonlinear Systems
77 Additional Functions, Conic Sections, and Nonlinear Systems Relations and functions are an essential part of mathematics as they allow to describe interactions between two or more variable quantities.
More informationModule 7 (Lecture 25) RETAINING WALLS
Module 7 (Lecture 25) RETAINING WALLS Topics Check for Bearing Capacity Failure Example Factor of Safety Against Overturning Factor of Safety Against Sliding Factor of Safety Against Bearing Capacity Failure
More informationMath 3 Unit 2: Solving Equations and Inequalities
Math 3 Unit 2: Solving Equations and Inequalities Unit Title Standards 2.1 Analyzing Piecewise Functions F.IF.9 2.2 Solve and Graph Absolute Value Equations F.IF.7B F.BF.3 2.3 Solve and Graph Absolute
More informationMath 301 Trigonometry Prac ce Exam 4. There are two op ons for PP( 5, mm), it can be drawn in SOLUTIONS
SOLUTIONS Math 0 Trigonometry Prac ce Exam Visit for more Math 0 Study Materials.. First determine quadrant terminates in. Since ssssss is nega ve in Quad III and IV, and tttttt is neg. in II and IV,
More information2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root
Academic Algebra II 1 st Semester Exam Mr. Pleacher Name I. Multiple Choice 1. Which is the solution of x 1 3x + 7? (A) x 4 (B) x 4 (C) x 4 (D) x 4. If the discriminant of a quadratic equation is zero,
More informationSecondary Math 2 Honors Unit 4 Graphing Quadratic Functions
SMH Secondary Math Honors Unit 4 Graphing Quadratic Functions 4.0 Forms of Quadratic Functions Form: ( ) f = a + b + c, where a 0. There are no parentheses. f = 3 + 7 Eample: ( ) Form: f ( ) = a( p)( q),
More informationRadicals and Radical Functions
0 Radicals and Radical Functions So far we have discussed polynomial and rational expressions and functions. In this chapter, we study algebraic expressions that contain radicals. For example, + 2, xx,
More informationSECTION 7: STEADYSTATE ERROR. ESE 499 Feedback Control Systems
SECTION 7: STEADYSTATE ERROR ESE 499 Feedback Control Systems 2 Introduction SteadyState Error Introduction 3 Consider a simple unityfeedback system The error is the difference between the reference
More information