Quadratic equations Revision Notes for Edexcel GCSE Maths

Quadratic equations

What are quadratic equations?

A quadratic equation is an equation where the highest power of the variable is 2. In Foundation GCSE, you need to solve quadratic equations without a calculator by factorising.

A quadratic equation in standard form looks like this: $x^2 + 6x + 5 = 0$

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The standard form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .

The key features are:

The left-hand side contains a quadratic expression
The right-hand side equals zero

The three-step method for solving quadratic equations

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The Three-Step Method for Factorising Quadratic Equations

Follow these three steps to solve any quadratic equation by factorising:

Step 1: Factorise the left-hand side of the equation

Look for two numbers that multiply to give the constant term and add to give the coefficient of x
Write the quadratic as two brackets: $(x + a)(x + b) = 0$

Step 2: Set each factor equal to zero

Use the zero product property: if two factors multiply to give zero, at least one of them must be zero
This gives you: $(x + a) = 0$ and $(x + b) = 0$

Step 3: Solve the two linear equations

Solve each simple equation to find the two solutions
Write both solutions clearly: $x = ?$ and $x = ?$

Worked examples

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Worked Example 1: Solving $x^2 + 8x - 9 = 0$

Step 1: Factorise the left-hand side

We need two numbers that multiply to give $-9$ and add to give $8$
These numbers are $+9$ and $-1$ : $(x + 9)(x - 1) = 0$

Step 2: Set each factor equal to zero

$x + 9 = 0$ and $x - 1 = 0$

Step 3: Solve the linear equations

$x = -9$ and $x = 1$

Check: $(1)^2 + 8(1) - 9 = 1 + 8 - 9 = 0$ ✓

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Worked Example 2: Solving $x^2 - 100 = 0$

Step 1: Factorise the left-hand side

This is a difference of two squares: $x^2 - 100 = (x + 10)(x - 10) = 0$

Step 2: Set each factor equal to zero

$x + 10 = 0$ and $x - 10 = 0$

Step 3: Solve the linear equations

$x = -10$ and $x = 10$

Special cases to watch out for

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When one solution is $x = 0$

For equations like $x^2 - 10x = 0$ :

Factor out $x$ : $x(x - 10) = 0$
Solutions are $x = 0$ and $x = 10$

This happens when there is no constant term in the quadratic equation.

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Difference of two squares

For equations like $x^2 - 25 = 0$ :

Recognise the pattern: $(x - 5)(x + 5) = 0$
Solutions are $x = 5$ and $x = -5$

The general form is $x^2 - a^2 = (x-a)(x+a)$

Alternative method using inverse operations

For simple quadratic equations like $x^2 - 100 = 0$ , you can also use inverse operations:

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Inverse Operations Method

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

Remember that there are two answers because $(-10)^2 = (-10) \times (-10) = 100$ .

Exam tips

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Essential Exam Tips

Always check your factorisation by expanding the brackets
Quadratic equations usually have two solutions
Write your final answers clearly: $x = ?$ and $x = ?$
Both solutions can be negative, positive, or one of each
Show all your working for full marks
Double-check by substituting your solutions back into the original equation

Summary

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Key Points to Remember:

Quadratic equations have $x^2$ as the highest power and equal zero
Use the three-step method: factorise, set equal to zero, solve
Zero product property: if $(x + a)(x + b) = 0$ , then $x + a = 0$ or $x + b = 0$
Most quadratic equations have two solutions
Always check your answers by substituting back into the original equation
Special cases include when $x = 0$ is a solution and difference of two squares patterns

Quadratic equations (Edexcel GCSE Maths): Revision Notes