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Factorising Factorising quadratics Expanding brackets Simplifying algebraic expressionsThis topic is relevant for:

Here we break down everything you need to know about solving equations. You’ll learn what linear and quadratic algebraic equations are, and how to solve all the different types of them.

At the end you’ll find solving equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

An equation is a **mathematical expression** that contains an **equals sign**.

E.g.

\[\begin{aligned}
y+6&=0\\
3(x-3)&=12\\
\frac{2x+2}{4}&=\frac{x-3}{3}\\
2x^{2}+3x-2&=0
\end{aligned}\]

There are two sides to an equation, with the left side being **equal** to the right side.

Equations will often involve algebra and contain unknowns (variables) which we often represent with letters such as

We can solve simple equations and more complicated equations to work out the value of these unknowns; they could involve fractions, decimals or integers.

Get your free solving equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free solving equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEIn order to solve equations, we need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

In GCSE Maths there are two main types of equations that we need to solve, both of which are covered below.

Within solving equations, you will find lessons on linear equations and quadratic equations.

Each method of solving equations is summarised below. For detailed examples, practice questions and worksheets on each one follow the links to the step by step guides.

There are 5 main types of linear equations we can solve.

Example of solving an equation with:

- One unknown

2Unknown on both sides

3With brackets

4With fractions

5Powers (exponents) and roots

We can check that our solution is correct by substituting it into the original equation.

**Step by step guide: Linear equations (coming soon)**

There are 4 main ways to solve quadratic equations.

Example of solving a quadratic equation by:

- Factorising

2Quadratic Formula

\[\begin{aligned}
x^{2}+4x-5&=0\\
\\
x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
\\
a&=1, \quad b=4, \quad c=-5\\
x&=\frac{-4\pm\sqrt{4^2-4(1)(-5)}}{2(1)}\\
\\
x&=1, \qquad x=-5\\
\end{aligned}\]

3Complete the square

\[\begin{aligned}
x^{2}+4x-5&=0\\
\\
(x+2)^{2}-9&=0\\
(x+2)^{2}&=9\\
x+2&=\pm\sqrt{9}\\
\\
x&=3-2\\
&=1\\
\\
x&=-3-2\\
&=-5
\end{aligned}\]

4Graphically

\[x^{2}+4x-5=0\]

The solutions/roots are found when the graph equals 0.

\[x=1,\qquad x=-5\]

We can check that our solution is correct by substituting it into the original equation.

**Step by step guide: Quadratic equations (coming soon)**

1. Solve: 4x − 2 = 14

Show answer

x = 4

2. Solve: 3x − 8 = x + 6

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x = 7

3. Solve: 3(x + 3) = 2(x − 2)

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x = − 13

4. Solve:

\[\frac{2 x+2}{3}=\frac{x-3}{2}\]

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x= − 13

5. Solve:

\[\frac{3 x^{2}}{2}=24\]

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\[x=\pm 4\]

6. Solve by factorising:

\[x^{2}-13 x+30=0\]

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\[(x – 3)(x – 10) = 0\]

\[x = 3, x = 10\]

7. Solve by factorising:

\[2 x^{2}+3 x-20=0\]

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\[(2x – 5)(x + 4) = 0\]

\[x=\frac{5}{2}, x=-4\]

8. (H) Solve by using the quadratic formula (give your answer to 3 significant figures):

\[x^{2}-4 x-3=0\]

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\[x=\frac{-(-4)+\sqrt{(-4)^{2}-4(1)(-3)}}{2(1)}\]

\[x=2+\sqrt{7}\]

\[x=4.65(3.s.f)\]

\[x=\frac{-(-4)-\sqrt{(-4)^{2}-4(1)(-3)}}{2(1)}\]

\[x=2-\sqrt{7}\]

\[x=-0.646(3.s.f)\]

9. (H) Solve by using the quadratic formula (give your answer to 3 significant figures):

\[2 x^{2}+4 x-4=0\]

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\[x=\frac{-(4)+\sqrt{(4)^{2}-4(2)(-4)}}{2(2)}\]

\[x=-1+\sqrt{3}\]

\[x=0.732(3.s.f)\]

\[x=\frac{-(4)+\sqrt{(4)^{2}-4(2)(-4)}}{2(2)}\]

\[x=-1-\sqrt{3}\]

\[x=-2.73(3.s.f)\]

10. (H) Solve by completing the square:

\[x^{2}-6 x+5=0\]

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\[(x-3)^{2}-4=0\]

\[x = 5, x = 1\]

1. Solve: 4y = 36

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y = 9

(1 mark)

2. Solve: x^{2} − 5x − 24 = 0

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x = − 3, x = 8

(3 marks)

3. Solve: 7y − 8 = 13

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y = 3

(2 marks)

- Use algebraic methods to solve linear equations
- Solve quadratic equations algebraically by factorising
- Solve quadratic equations algebraically by completing the square (H)
- Solve quadratic equations algebraically by using the quadratic formula (H)
- Solve quadratic equations by finding approximate solutions using a graph

- Linear equations
- Quadratic equations
- Quadratic equations (factorising)
- Quadratic formula
- Completing the square
- Simultaneous equations

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