Solving Quadratic Equations: Factorising, Formula and Completing the Square

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 11 · Equations and Inequalities

MEMORANDUM

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Class:

Date:

Mark

/ 36

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Solve x² − 7x + 12 = 0. (factorise)
(2)
(x − 3)(x − 4)=0 → x = 3 or 4✓✓ (2)
2
Solve x² + 2x − 8 = 0. (factorise)
(2)
(x + 4)(x − 2)=0 → x = −4 or 2✓✓ (2)
3
Solve x² − 9x = 0. (common factor)
(2)
x(x − 9)=0 → x = 0 or 9✓✓ (2)
4
Solve x² − 16 = 0. (difference of squares)
(2)
(x − 4)(x + 4)=0 → x = ±4✓✓ (2)
5
Solve x² + 4x + 1 = 0 (formula, 2 decimal places). (formula)
(3)
x = (−4±√12)/2 ≈ −0,27 or −3,73✓✓✓ (3)
6
Solve 2x² − 5x + 1 = 0 (formula, 2 decimal places). (formula)
(3)
x = (5±√17)/4 ≈ 2,28 or 0,22✓✓✓ (3)
7
Solve x² + 8x − 9 = 0 by completing the square. (complete the square)
(3)
x² + 8x = 9; half of 8 is 4 and 4² = 16, so x² + 8x + 16 = 9 + 16 = 25; (x + 4)² = 25; x + 4 = ±5 → x = 1 or x = −9✓✓✓ (3)
8
Solve 3x² − 12x + 5 = 0 by completing the square, correct to TWO decimal places. (complete the square)
(4)
÷ 3: x² − 4x + 5/3 = 0 → x² − 4x = −5/3; add 4: (x − 2)² = 7/3; x − 2 = ±1,5275 → x ≈ 3,53 or x ≈ 0,47✓✓✓✓ (4)
9
Write x² − 10x + 3 in the form (x + p)² + q. (complete the square)
(2)
Half of −10 is −5 and (−5)² = 25, so x² − 10x + 3 = (x − 5)² − 25 + 3 = (x − 5)² − 22✓✓ (2)
10
Solve x² = 2x + 15. (rearrange)
(3)
x² − 2x − 15=0 → (x − 5)(x + 3)=0 → x = 5 or −3✓✓✓ (3)
11
Solve 2x² + 7x + 3 = 0. (leading coeff)
(3)
(2x + 1)(x + 3)=0 → x = −1/2 or −3✓✓✓ (3)
12
Solve x + 5/x = 6. (fraction)
(4)
x² − 6x + 5=0 → (x − 1)(x − 5)=0 → x = 1 or 5✓✓✓✓ (4)
13
Solve (x − 2)(x + 1) = 4. (expand first)
(3)
x² − x − 2=4 → x² − x − 6=0 → (x − 3)(x + 2)=0 → x = 3 or −2✓✓✓ (3)

TOTAL: 36 marks