# Solving a Quadratic Equation

(KS4, Year 10)

**x**that makes this equation true (i.e. makes the left hand side equal to 0.) The values of

**x**that solve the equation are called the

*roots*of the equation.

## Understanding Solving Quadratic Equations

It is easier to understand solving quadratic equations with an example. Let's look at a quadratic equation.**x**is a variable. It can take different values. Let's try

**x = 1**,

**x = 2**and

**x = 3**.

## x = 1

Substitute**x = 1**into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*1* )^{2} − 3(*1* ) + 2

x^{2} − 3x + 2 = *1* × *1* − 3 × *1* + 2

x^{2} − 3x + 2 = 1 − 3 + 2

x^{2} − 3x + 2 = 0

**x = 1**, the left hand side of the equation equals

**0**, which is equal to the right hand side of the equation.

**x = 1**solves the equation. It is a root of the equation.

## x = 2

Substitute**x = 2**into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*2* )^{2} − 3(*2* ) + 2

x^{2} − 3x + 2 = *2* × *2* − 3 × *2* + 2

x^{2} − 3x + 2 = 4 − 6 + 2

x^{2} − 3x + 2 = 0

**x = 2**, both sides of the equation are equal.

**x = 2**solves the equation. It is a root of the equation.

## x = 3

Substitute**x = 3**into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*3* )^{2} − 3(*3* ) + 2

x^{2} − 3x + 2 = *3* × *3* − 3 × *3* + 2

x^{2} − 3x + 2 = 9 − 9 + 2

x^{2} − 3x + 2 = 2 ≠ 0

**x = 3**, the left hand side of the equation equals

**2**. This is

*not*equal to the right hand side of the equation,

**0**.

**x = 3**does not solve the equation.

**x = 1**and

**x = 2**solve the quadratic equation

**x**. A quadratic equation will always have

^{2}− 3x + 2 = 0**2**values of

**x**that solve the equation. There are always

**2**roots.

## How to Solve Quadratic Equations

There are 3 ways to solve quadratic equations.## (1) Factoring

A quadratic equation can sometimes be written as the product of two brackets. For example: from this, we can read off the two roots of the quadratic equation:**x = 1**,

**x = 2**

solving quadratic equations using factoring

## (2) Quadratic Formula

A quadratic equation can be solved using the quadratic formula: In this formula,**a**,

**b**and

**c**are the numbers in the quadratic equation in standard form,

**ax**.

^{2}+ bx + csolving quadratic equations using the quadratic formula

## (3) Graph

A quadratic equation can be solved by plotting it on a graph and finding where it crosses the x-axis: In this graph above, the quadratic curve crosses the x-axis at**x = 1**and

**x = 2**. These are the roots of the equation, that solve the equation.

solving quadratic equations using a graph

## What's in a Name?

The word "quadratic" comes from the word "quad", meaning "square" - because the**x**is squared.

## Factoring, Factorising

To write a quadratic equation as a product of two brackets is called 'to factor' or 'to factorise' the quadratic equation. The method is refered to as 'factoring' or 'factorising'.## There Are 2 Roots

Quadratic equations always have two solutions. There are 2 values of**x**that solve the equation. We can visualize this by looking at a graph of a quadratic equation. The roots are the points where the curve crosses the horizontal x-axis.

- There can be two distinct roots. We see this because the curve crosses the x-axis at 2 separate places:
- Sometimes it seems that there is only one root. But that root is repeated.
- Even when it seems there are no roots, there are two complex roots.

## Worksheet

This test is printable and sendable