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1. Factorise: 2 - - 6 2. Solve for : 2( + 1) = - 1 3.

State the gradient of the line: + 12 = 2 Apr 26 17:3 Apr 26 18:27 Solving Quadratic

factorising. Eamples: 1) 2-2 - 35 = 0 2) 2 2 + 10 = 0 1. Simplif 2. Simplif 3. Factorise.

1 1. Factorise: Solve for : 2( + 1) = Factorise: 2-25 To solve quadratic equations.. Factorise: State the gradient of the line: + 12 = 2 Apr 26 17:3 Apr 26 18:27 Solving Quadratic Equations A quadratic equation can be written as a 2 + b + c = 0 Then, we can solve b factorising. Eamples: 1) = 0 2) = 0 1. Simplif 2. Simplif 3. Factorise. Solve the inequalit: 5. Find Apr 26 18:0 Apr 28 10:22 To solve quadratic equations where a 1. Eamples: 1) Solving Quadratic Equations 2) Apr 28 10:33 Apr 28 10:3 1

2 Finding a Formula The Quadratic Formula If we have an equation a 2 + b + c = 0 that we can't factorise, we can use the Quadratic Formula to find solutions: Eamples: 1) (given in eams) 2) Ma 1 16:53 Ma 1 17:03 1. Simplif 2. Find the length of the arc: 128 o 20m Practising solving equations using the quadratic formula. 3. Simplif. Find the gradient of the straight line that joins the points (3, ) and (7, 10). Ma 3 17: Ma 3 17:5 a 2 + b + c = 0 Solve the following equation to 2 decimal places: 1. Change the subject of the formula to a: (a + b) 2 = c 2. Evaluate 3. Epress with a rational denominator.. Evaluate Ma 3 17:6 Ma 09:05 2

The area is 30cm 2. Find p. p 5cm. Evaluate 16 3 Ma 09:05 Ma 7 18:0 Solve to 2 decimal places: Solve to 2 decimal places: = -0.

3 - No Calculators! (a - b) + b2 1. If a = 3 and b = -, evaluate 11 To practise using the quadratic formula to solve more difficult equations. 2. Simplif: 3. The area is 30cm 2. Find p. p 5cm. Evaluate 16 3 Ma 09:05 Ma 7 18:0 Solve to 2 decimal places: Solve to 2 decimal places: = -0.6 or -2.0 = 3.73 or 0.27 Ma 7 18:0 Ma 7 18:0 Solve to 2 decimal places: 1) Find : 15cm 17cm 2) Factorise the epression: = 0.26 or ) Calculate ) List all the prime numbers that are smaller than 20 Ma 7 18:0 Ma 9 18:57 3

4 The areas of these rectangles are equal. a) Find the value of. b) Calculate the area of the rectangles. To be able to solve problems using quadratic equations. ( + 1) cm ( + 3) cm (2 + 2) cm ( + ) cm Ma 9 18:58 Ma 9 18:58 1) Solve the following equations: a) = 0 b) = 0 Considering quadratic graphs and their features. 2) Solve the equation to 1 decimal place: = 0 Ma 11 15:09 Ma 11 15:3 What are the features of a quadratic graph? Graph of = k 2 = = Eample: positive k negative k Find k from the graph of = = k 2 : (2, 16) = = Ma 11 15:18 Ma 11 15:3

5 1) If a = -3, evaluate: -2a 2 + 3a 2) Simplif: ) Find b: b 35 o 10cm ) What is 0ml increased b 20%? Ma 12 1:0 Ma 12 1:3 What is the equation of this graph? To recognise the graph of = k 2 + q and be able to find k and q from a graph. Ma 12 1:6 Ma 12 1:5 The graph of = 2 + q Find the equations of these graphs, of the form = k 2 + q 1) 2) 3) positive q negative q Ma 12 1:56 Ma 12 15:17 5

6 1) If these rectangles have the same area, find : m (2-2) m 5m 2 m This quadratic graph has been shifted to the right. Can ou find its equation? 2) Simplif a b -1 ab 5 3) Find a: 55 o a ) Calculate cm Ma 1 18:21 Ma 1 18:31 The graph of = ( + p) 2 1) Find p for these graphs of = ( + p) 2 : a) b) c) (2,16) (6,0) (-5,0) positive p negative p Ma 1 18:3 Ma 1 18:36 1) Find a and b, given: 2a - b = 2 a + b = 7 2) Calculate , giving our answer in scientific notation To find p and q from the graph of = ( +p) 2 + q. 3) Make g the subject of the fomula T = 5 - L g ) Round to 3 sig. fig Ma 16 17:7 Ma 16 18:1 6

Eample: The graph of = ( +p) 2 + q (-3,2) For each of these graphs of the form = ( +p) 2 + q, find p and q: 1) 2) 3) ) (3,2) (,-3) (-1,-) (-1,2) Ma 16 18:20 Ma 16 18:18 1) Solve for : 2 + 5-2 = 0 2)

3 ) Rearrange for h: a = h + 5 h = 3-5a a Ma 18 12:01 Ma 18 12:03 Sketching Quadratic Graphs Sketches of quadratic graphs can involve finding: roots (where graph cuts the -ais) the -intercept the

7 Eample: The graph of = ( +p) 2 + q (-3,2) For each of these graphs of the form = ( +p) 2 + q, find p and q: 1) 2) 3) ) (3,2) (,-3) (-1,-) (-1,2) Ma 16 18:20 Ma 16 18:18 1) Solve for : = 0 2) Find the area of the sector: 12cm 77.9cm 2 62 o Sketch the graph of = ( - 3) 2-3) Round to sig. fig. 3 ) Rearrange for h: a = h + 5 h = 3-5a a Ma 18 12:01 Ma 18 12:03 Sketching Quadratic Graphs Sketches of quadratic graphs can involve finding: roots (where graph cuts the -ais) the -intercept the equation of the line of smmetr the coodinates of the turning point and its nature 1) Simplif 2) Calculate ) Calculate the volume of the clinder: 6cm 8cm 72π Eamples 1) Sketch the graph of = ( + 1) ) A car was bought two ears ago and has since lost 30% of its value. If its current value is 6300 then what was its original value? Ma 18 12:08 Ma 19 13:50 7

8 = ( - 1) 2-25 Roots: 0 = = ( - )( + 6) = or -6 intercept: = (-1) 2-25 = -26 Spot the mistake(s)! Equation of ais of smmetr: = -1 (-6,0) TP occurs at (-1, -25) and is a minimum because 2 >0 (,0) (0,-26) (-1,-25) To sketch the graph of -( + p) 2 + q. Ma 19 13:5 Ma 19 13:51 2) Sketch the graph of = -( + 1) 2 + To sketch graphs of the form = (a + b)(c + d). Ma 19 13:52 Ma 22 16:00 Sketching = (a + b)(c + d) 1) Write in completed square form: m 2-8m - 2 -t g = s - 3 2) Change the subject of the fomula to g: s = 3 - t g 3) Find the highest common factor of m n 3 and 12 mn 5 E.g. Sketch the graph = ( + 2)( - ) ) Calculate (without a calculator) Ma 22 16:02 Ma 25 18:3 8

9 The Discriminant For a quadratic equation a 2 + b + c the discriminant is b 2 - ac. Match the equations to the graphs b finding the roots. How can we tell a graph onl has one root (or no roots at all) without sketching it? b 2 - ac > 0 means 2 real, distinct roots b 2 - ac = 0 means 2 real, equal roots b 2 - ac < 0 means no real roots e.g. Determine the nature of the roots of 2( + 1) = 2-3 Ma 25 18:3 Ma 25 19:12 : The Discriminant 1) Determine the nature of the roots of: The discriminant is b 2 - ac. a) 0 = b) = 0 Find the range of values of k such that k = 0 has real roots. c) (2-1) 2-3( + 1) = 0 d) ( + 3) = 2-3 Ma 26 1: Ma 26 17:50 To use what we have learnt to solve problems. Ma 26 1:7 Ma 26 18:07 9

12. Quadratics NOTES.notebook September 21, 2017

12. Quadratics NOTES.notebook September 21, 2017 1) Fully factorise 4y 2-5y - 6 Today's Learning: To find the equation of quadratic graphs using substitution of a point. 2) Epand the brackets and simplify: (m + 4)(2m - 3) 3) Calculate 20% of 340 without