Solving Quadratic Equations

A quadratic equation contains an x^2 term as the highest power and has the general form ax^2 + bx + c = 0. Unlike linear equations, quadratic equations can have two solutions (roots). This topic is tested heavily on both Foundation and Higher tier AQA GCSE Mathematics papers. Foundation students need to solve by factorising, while Higher students must also use the quadratic formula and complete the square.

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Solving by Factorising

This is the most common method and is required at all levels. To solve a quadratic by factorising:

1. Rearrange the equation so that one side equals zero.
2. Factorise the quadratic expression.
3. Set each bracket equal to zero and solve.

Worked Example 1

Solve: x^2 + 5x + 6 = 0

Step 1: Factorise. Find two numbers that multiply to 6 and add to 5: 2 and 3.

x^2 + 5x + 6 = (x + 2)(x + 3) = 0

Step 2: Set each bracket equal to zero:

x + 2 = 0, so x = -2

x + 3 = 0, so x = -3

Answer: x = -2 or x = -3

Worked Example 2

Solve: x^2 - 3x - 10 = 0

Find two numbers that multiply to -10 and add to -3: -5 and +2.

(x - 5)(x + 2) = 0

x = 5 or x = -2

Worked Example 3

Solve: x^2 - 9 = 0

This is a difference of two squares: (x + 3)(x - 3) = 0

x = 3 or x = -3

Exam Tip: Always rearrange the equation to set one side equal to zero before factorising. If you try to factorise without doing this first, the method will not work.

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Solving When the Equation is Not Already Rearranged

Sometimes the equation is not given in the form ax^2 + bx + c = 0. You need to rearrange first.

Worked Example 4

Solve: x^2 + 3x = 28

Rearrange: x^2 + 3x - 28 = 0

Find two numbers that multiply to -28 and add to 3: 7 and -4.

(x + 7)(x - 4) = 0

x = -7 or x = 4

Worked Example 5

Solve: 2x^2 = 18

Divide both sides by 2: x^2 = 9

Take the square root of both sides: x = 3 or x = -3

Exam Tip: When you take the square root of both sides of an equation, always remember to include both the positive and negative roots. Writing only x = 3 instead of x = 3 or x = -3 will lose you a mark.

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Factorising When a is Not 1 [H]

For quadratics like 2x^2 + 7x + 3 = 0, use the ac method from the previous lesson.

Worked Example 6

Solve: 2x^2 + 7x + 3 = 0

a x c = 2 x 3 = 6. Find two numbers that multiply to 6 and add to 7: 6 and 1.

Rewrite: 2x^2 + 6x + x + 3 = 0

Group: 2x(x + 3) + 1(x + 3) = 0

Factorise: (2x + 1)(x + 3) = 0

2x + 1 = 0 gives x = -1/2

x + 3 = 0 gives x = -3

Answer: x = -1/2 or x = -3

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The Quadratic Formula [H]

When a quadratic cannot be factorised easily, use the quadratic formula. For the equation ax^2 + bx + c = 0:

x = (-b plus or minus the square root of (b^2 - 4ac)) / 2a

The expression b^2 - 4ac is called the discriminant.

Discriminant Value	Meaning
b^2 - 4ac > 0	Two distinct real solutions
b^2 - 4ac = 0	One repeated real solution
b^2 - 4ac < 0	No real solutions

Worked Example 7

Solve: 3x^2 + 2x - 5 = 0, giving answers to 2 decimal places.

Here a = 3, b = 2, c = -5.

Discriminant: b^2 - 4ac = (2)^2 - 4(3)(-5) = 4 + 60 = 64

Square root of 64 = 8

x = (-2 + 8) / (2 x 3) = 6/6 = 1

x = (-2 - 8) / (2 x 3) = -10/6 = -1.67 (2 d.p.)

Worked Example 8

Solve: x^2 + 3x - 1 = 0, giving answers to 2 decimal places.

a = 1, b = 3, c = -1.

Discriminant: 9 - 4(1)(-1) = 9 + 4 = 13

Square root of 13 = 3.6056...

x = (-3 + 3.6056) / 2 = 0.6056 / 2 = 0.30 (2 d.p.)

x = (-3 - 3.6056) / 2 = -6.6056 / 2 = -3.30 (2 d.p.)

Exam Tip: The quadratic formula is given on the AQA formula sheet, so you do not need to memorise it. However, practise using it so that you can substitute values accurately under exam pressure. Be especially careful with negative values of b and c.

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Completing the Square [H]

Completing the square rewrites a quadratic in the form (x + p)^2 + q. This method is useful for finding the minimum or maximum point of a parabola and for solving quadratics.

The Method

For x^2 + bx + c:

1. Halve the coefficient of x: p = b/2
2. Write (x + p)^2
3. Subtract p^2 and add c: (x + p)^2 - p^2 + c

Worked Example 9

Write x^2 + 6x + 2 in completed square form.

Step 1: Halve the coefficient of x: 6/2 = 3

Step 2: Write (x + 3)^2

Step 3: Subtract 3^2 = 9, add the constant 2: (x + 3)^2 - 9 + 2 = (x + 3)^2 - 7

Worked Example 10

Solve x^2 + 6x + 2 = 0 by completing the square.

From above: (x + 3)^2 - 7 = 0

(x + 3)^2 = 7

x + 3 = plus or minus the square root of 7

x = -3 + square root of 7 = -0.35 (2 d.p.)

x = -3 - square root of 7 = -5.65 (2 d.p.)

flowchart TD
    A[Quadratic equation ax^2 + bx + c = 0] --> B{Can it be factorised easily?}
    B -- Yes --> C[Factorise and set each bracket = 0]
    B -- No --> D{Do you need exact answers or the turning point?}
    D -- "Decimal answers" --> E[Use the quadratic formula]
    D -- "Exact/turning point" --> F[Complete the square]

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Setting Up Quadratics from Problems

Some questions give a real-life context and require you to form a quadratic equation before solving.

Worked Example 11

A rectangular garden has a length that is 4 metres more than its width. The area of the garden is 60 m^2. Find the dimensions.