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5.8H Solving Quadratics: Higher Tier Practice

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Factorise fully: x² + 5x - 14.

(Total for Question 1 is 2 marks)

Solve the equation 2x² - x - 6 = 0.

(Total for Question 2 is 3 marks)

Show that x² - 6x + 7 can be written in the form (x - 3)² - 2. Give full algebraic working.

(Total for Question 3 is 3 marks)

Solve 3x² + 2x - 5 = 0 giving exact answers in surd form.

(Total for Question 4 is 4 marks)

Find the value of k such that the quadratic x² + (k - 2)x + k has equal roots.

(Total for Question 5 is 3 marks)

A rectangle has sides of length x + 4 and x + 2. The area is 168 cm². Find the dimensions of the rectangle (give the positive solution for x).

(Total for Question 6 is 3 marks)

Solve x² = 2x + 3.

(Total for Question 7 is 3 marks)

For which values of m does the quadratic f(x) = x² - 4x + m have two distinct real positive roots? Show your reasoning.

(Total for Question 8 is 5 marks)

Solve 1/2 x² - 2x + 3/2 = 0. Give exact answers.

(Total for Question 9 is 3 marks)

Find the value of k such that x² + kx - 12 is divisible by x - 3.

(Total for Question 10 is 2 marks)

Find the vertex (turning point) and minimum value of y = 2x² - 8x + 5.

(Total for Question 11 is 3 marks)

Solve 2x² - 7x = x + 3 and give exact answers in simplest form.

(Total for Question 12 is 4 marks)

Mark scheme · 5.8H Solving Quadratics: Higher Tier Practice

Question 1

M1 identify two numbers multiplying to -14 and summing to 5
A1 (x + 7)(x - 2) cao
Answer: (x + 7)(x - 2)

Question 2

M1 correct factorisation or rearrangement shown, e.g. 2x² +3x -4x -6 or (2x +3)(x -2)
M1 obtain factors (2x +3)(x -2) oe
A1 x = 2 or x = -3/2 cao
Answer: x = 2 or x = -3/2

Question 3

M1 write x² - 6x + 7 as x² - 6x + 9 - 9 + 7 or equivalent completing-the-square step
M1 group to (x - 3)² - 2 or show algebraic rearrangement
A1 (x - 3)² - 2 cso
Answer: (x - 3)² - 2

Question 4

M1 substitute into quadratic formula x = [-b ± √b² - 4ac]/(2a) with a=3, b=2, c=-5
M1 evaluate discriminant: b² - 4ac = 4 + 60 = 64
A1 √64 = 8 substituted correctly
A1 x = (-2 ± 8) / 6 giving x = 1 or x = -5/3 cao
Answer: x = 1 or x = -5/3

Question 5

M1 state discriminant = 0 for equal roots and form (k - 2)² - 4*1*k = 0 or b² - 4ac = 0
M1 expand and simplify: k² - 4k + 4 - 4k = 0 giving k² - 8k + 4 = 0
A1 solve k = [8 ± √64 - 16]/2 = [8 ± √48]/2 = 4 ± √12 oe; both values (k = 4 + 2sqrt(3) or k = 4 - 2sqrt(3)) cao
Answer: k = 4 + 2sqrt(3) or k = 4 - 2sqrt(3)

Question 6

M1 form equation (x + 4)(x + 2) = 168 and expand to x² + 6x + 8 = 168
M1 rearrange to x² + 6x - 160 = 0 and solve by factorisation or formula to get x = 10 or x = -16
A1 positive solution x = 10 giving dimensions 14 cm and 12 cm cao
Answer: Dimensions are 14 cm by 12 cm (x = 10)

Question 7

M1 rearrange to x² - 2x - 3 = 0
M1 factorise to (x - 3)(x + 1) oe
A1 x = 3 or x = -1 cao
Answer: x = 3 or x = -1

Question 8

M1 state discriminant > 0 and compute: b² - 4ac = 16 - 4m > 0 leading to m < 4
M1 state product of roots = c/a = m must be > 0 for roots both positive
M1 note sum of roots = 4 which is > 0 so does not restrict further (state sum > 0)
A1 combine conditions to 0 < m < 4 and justify distinctness by strict inequalities
A1 final answer 0 < m < 4 cao
Answer: 0 < m < 4

Question 9

M1 multiply both sides by 2 to get x² - 4x + 3 = 0 or use quadratic formula
M1 factorise or solve to get (x - 1)(x - 3) = 0
A1 x = 1 or x = 3 cao
Answer: x = 1 or x = 3

Question 10

M1 use remainder theorem: substitute x = 3 into polynomial to get 9 + 3k - 12 = 0
A1 solve 3k - 3 = 0 giving k = 1 cao
Answer: k = 1

Question 11

M1 complete the square: 2(x² - 4x) + 5 = 2[(x - 2)² - 4] + 5 or equivalent
M1 simplify to 2(x - 2)² - 3
A1 vertex (2, -3) and minimum value -3 cao
Answer: Turning point at (2, -3); minimum value -3

Question 12

M1 bring all terms to one side: 2x² - 8x - 3 = 0
M1 substitute into quadratic formula or show work: x = [8 ± √64 + 24]/4
A1 simplify under root to √88 = 2sqrt(22)
A1 final answers x = (4 ± √22)/2 cao
Answer: x = (4 + √22)/2 or x = (4 - √22)/2

Solving Quadratics: Higher Tier Practice - Worksheets, Questions and Revision