# Worksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra

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1 Worksheets for GCSE Mathematics Quadratics mr-mathematics.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, y-intercept 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

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