Quadratic equations

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Solving by factorising

A quadratic equation has form ax² + bx + c = 0 with a ≠ 0. Rearrange one side to zero before using the zero-product method.
Factorise and use the fact that if AB = 0 then A = 0 or B = 0. x² - 5x + 6 = 0 gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
For ax² + bx + c with a \neq 1, find two numbers that multiply to ac and add to b, then split the middle term.
x² - 49 = 0 is a difference of two squares: (x - 7)(x + 7) = 0.

The quadratic formula

When a quadratic will not factorise easily, use x = \frac{-b \pm \√{b² - 4ac}}{2a}.
Substitute a, b and c carefully, using brackets for negative values.
The part under the root, b² - 4ac, is the discriminant: positive gives two roots, zero gives one repeated root, negative gives no real roots.

Completing the square

Write x² + bx + c as (x + \frac{b}{2})² + c - (\frac{b}{2})².
x² + 6x + 2 = (x + 3)² − 7. For the equation x² + 6x + 2 = 0, this gives (x + 3)² = 7 and x = −3 ± √7.
The turning point of (x + p)² + q is (-p, q).

Common errors

Not setting the equation equal to zero first (x² + 3x = 10 is not x(x + 3) = 10 \Rightarrow x = 10).
Losing the second solution when taking a square root: x² = 9 gives x = \pm 3.
Sign slips when b is negative in the formula: -b becomes positive.

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Practice questions

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1.What must a quadratic equation equal before you factorise it to solve?
Easy
- A1
- B0
- Cx
- Dc
2.Solve x² - 5x + 6 = 0.
Easy
- Ax = 2, 3
- Bx = -2, -3
- Cx = 1, 6
- Dx = -1, -6
3.Solve x² = 25.
Easy
- Ax = 5
- Bx = \pm 5
- Cx = 12.5
- Dx = \pm 12.5
4.Solve x² + 7x + 10 = 0.
Easy
- Ax = 2, 5
- Bx = -2, -5
- Cx = 1, 10
- Dx = -1, -10
5.Solve x² - x - 6 = 0.
Easy
- Ax = 3, -2
- Bx = -3, 2
- Cx = 6, -1
- Dx = 1, -6
6.Solve x² - 49 = 0.
Easy
- Ax = 7
- Bx = \pm 7
- Cx = \pm 49
- Dx = 24.5
7.Solve x² + 4x = 0.
Easy
- Ax = 0, -4
- Bx = 4
- Cx = 0, 4
- Dx = -4
8.Solve x² + 3x = 10.
Medium
- Ax = 2, -5
- Bx = -2, 5
- Cx = 3, 7
- Dx = 10, -1

More questions

9.Solve 2x² + 5x + 2 = 0.
Medium
- Ax = -2, -\frac{1}{2}
- Bx = 2, \frac{1}{2}
- Cx = -1, -4
- Dx = 1, 4
10.Solve 3x² - 7x + 2 = 0.
Medium
- Ax = 2, \frac{1}{3}
- Bx = -2, -\frac{1}{3}
- Cx = 3, \frac{2}{3}
- Dx = 1, 6
11.What is the discriminant of x² - 6x + 9 = 0?
Medium
- A0
- B36
- C72
- D-36
12.What is the discriminant of x² + 3x + 4 = 0?
Medium
- A25
- B-7
- C7
- D9
13.What does a discriminant of -7 tell you?
Medium
- Atwo real roots
- Bone real root
- Cno real roots
- Dthree roots
14.Complete the square: x² + 6x + 2.
Medium
- A(x + 3)² - 7
- B(x + 3)² + 2
- C(x + 6)² - 34
- D(x + 3)² - 9
15.Complete the square: x² - 8x + 3.
Medium
- A(x - 4)² - 13
- B(x - 4)² + 3
- C(x - 8)² - 61
- D(x - 4)² + 13
16.Solve x² + 4x - 3 = 0 giving answers to 2 decimal places.
Medium
- Ax = 0.65, -4.65
- Bx = 1, -3
- Cx = 0.46, -4.46
- Dx = 3.65, -0.65
17.Use the formula to solve 2x² - 3x - 4 = 0 (2 d.p.).
Medium
- Ax = 2.35, -0.85
- Bx = 1.5, -4
- Cx = 2.5, -0.5
- Dx = 3, -1
18.The roots of x² + bx + 12 = 0 are 3 and 4. What is b?
Medium
- A-7
- B7
- C12
- D-12
19.The area of a rectangle is 24 cm². The length is 2 cm more than the width x. Which equation gives x?
Medium
- Ax² + 2x - 24 = 0
- Bx² - 2x - 24 = 0
- C2x² = 24
- Dx + 2 = 24
20.For the rectangle x(x + 2) = 24, what is x?
Medium
- A4
- B6
- C-6
- D22
21.x² - 6x + 11 written as (x - p)² + q has q equal to:
Medium
- A2
- B11
- C-2
- D20
22.Why must you consider both + and - when solving x² = 36?
Easy
- Abecause (-6)² = 36 too
- Bbecause 36 is even
- Cto make the answer integer
- Dthere is no reason
23.When is the quadratic formula most useful?
Medium
- Awhen the quadratic factorises easily
- Bwhen it will not factorise easily
- Cwhen there is no x²
- Dfor linear equations
24.Which is the correct quadratic formula?
Easy
- A\frac{b \pm \√{b² - 4ac}}{2a}
- B\frac{-b \pm \√{b² - 4ac}}{2a}
- C\frac{-b \pm \√{b² + 4ac}}{2a}
- D\frac{-b \pm \√{b² - 4ac}}{a}
25.x² - 4x + k = 0 has one repeated root. What is k?
Medium
- A2
- B4
- C8
- D-4
26.Solving x² = 49 gives x = 7 only.
Easy
True or false?
27.A quadratic equation can have two different solutions.
Easy
True or false?
28.If the discriminant is negative, the quadratic has no real roots.
Easy
True or false?
29.x² + 3x = 10 can be solved by writing x(x + 3) = 10 so x = 10.
Easy
True or false?
30.The turning point of (x + 2)² - 5 is (2, -5).
Medium
True or false?
31.Which equations have solutions x = 2 and x = 3?
Medium
- Ax² - 5x + 6 = 0
- B(x - 2)(x - 3) = 0
- Cx² + 5x + 6 = 0
- Dx² - x - 6 = 0
- E2x² - 10x + 12 = 0
32.Which have no real solutions?
Medium
- Ax² + 1 = 0
- Bx² - 1 = 0
- Cx² + 2x + 3 = 0
- Dx² + 2x - 3 = 0
- Ex² = 0
33.Solve x² + 6x + 4 = 0 by completing the square, giving your answers in surd form.
Medium

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