Quadratic equations Revision Notes for AQA GCSE Maths

Quadratic equations

What are quadratic equations?

A quadratic equation is an equation where the highest power of x is 2. In your Foundation exam, you'll need to solve these equations without a calculator using factorisation.

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Quadratic equations follow this pattern:

The left-hand side contains a quadratic expression (like $x^2 + 6x + 5$ )
The right-hand side equals zero

Solving quadratic equations by factorisation

The factorisation method involves three clear steps that you must follow in order.

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The Three-Step Method for Solving Quadratic Equations:

Factorise the left-hand side
Set each factor equal to zero
Solve the linear equations

Step-by-step method

Step 1: Factorise the left-hand side

Rewrite the quadratic expression as two brackets multiplied together
For example: $x^2 + 6x + 5$ becomes $(x + 5)(x + 1)$

Step 2: Set each factor equal to zero

Since the right-hand side equals zero, at least one bracket must equal zero
Write out: $(x + 5) = 0$ and $(x + 1) = 0$

Step 3: Solve the linear equations

Solve each simple equation to find your two solutions
From the example: $x = -5$ and $x = -1$
Always write your final answers in the form $x = \text{[number]}$

Worked examples

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Worked Example 1: Standard Quadratic

For $x^2 + 8x - 9 = 0$ :

Step 1: Factorise to get $(x + 9)(x - 1) = 0$

Step 2: Set each factor to zero: $x + 9 = 0$ and $x - 1 = 0$

Step 3: Solutions are $x = -9$ and $x = 1$

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Worked Example 2: Difference of Two Squares

For $x^2 - 100 = 0$ :

Method 1 (Factorisation):

Recognise this as a difference of two squares
Factorise to get $(x + 10)(x - 10) = 0$
Solutions are $x = -10$ and $x = 10$

Method 2 (Inverse Operations):

Add 100 to both sides: $x^2 = 100$
Take the square root: $x = \pm 10$

Special cases to watch

When one solution is x = 0

If you see an equation like $x^2 - 10x = 0$ , this represents a special case where you can factor out $x$ .

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Solving equations with x = 0 as a solution:

Factor out $x$ to get $x(x - 10) = 0$
One solution will always be $x = 0$
The other solution comes from the second bracket: $x = 10$

Difference of two squares

Look for equations in the form $x^2 - a^2 = 0$ where both terms are perfect squares.

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Difference of Two Squares Pattern:

These factorise as $(x + a)(x - a) = 0$
Example: $x^2 - 25 = 0$ becomes $(x + 5)(x - 5) = 0$
Solutions are $x = 5$ and $x = -5$

Checking your solutions

Always verify your answers by substituting back into the original equation. Both solutions should make the left-hand side equal zero.

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How to check your solutions:

Take each solution and substitute it back into the original equation
Calculate the left-hand side
Confirm it equals zero
If it doesn't equal zero, check your working for errors

Exam tips

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Essential Exam Strategy:

Questions are typically worth 2-3 marks
Show all your working clearly - this is where most marks come from
Write your final answers in the form $x = \text{[number]}$
Look out for special cases that might have simpler methods
Both solutions are often negative in exam questions

Remember!

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Key Points to Remember:

Quadratic equations have $x^2$ as the highest power and equal zero on one side
Follow the three-step method: factorise, set each factor to zero, solve the linear equations
Always write solutions in the form $x = \text{[number]}$
Special cases include when one solution is $x = 0$ and difference of two squares patterns
Check your answers by substituting back into the original equation

Quadratic equations (AQA GCSE Maths): Revision Notes