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Solving a Quadratic Equation

(KS4, Year 10)

A quadratic equation is an equation in the form: y equals a x squared plus b x plus c

Solving a quadratic equation mean finding the value of 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:

x² − 3x + 2 = (1 )² − 3(1 ) + 2

x² − 3x + 2 = 1 × 1 − 3 × 1 + 2

x² − 3x + 2 = 1 − 3 + 2

x² − 3x + 2 = 0

When 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:

x² − 3x + 2 = (2 )² − 3(2 ) + 2

x² − 3x + 2 = 2 × 2 − 3 × 2 + 2

x² − 3x + 2 = 4 − 6 + 2

x² − 3x + 2 = 0

When 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:

x² − 3x + 2 = (3 )² − 3(3 ) + 2

x² − 3x + 2 = 3 × 3 − 3 × 3 + 2

x² − 3x + 2 = 9 − 9 + 2

x² − 3x + 2 = 2 ≠ 0

When 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 x² − 3x + 2 = 0. A quadratic equation will always have 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: x squared minus 3 x plus 2 equals (x minus 1) (x minus 2)

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: x equals minus b plus or minus the square root of b squared minus 4 a c all divided by 2 a

x equals minus b plus or minus the square root of b squared minus 4 a c all divided by 2 a

In this formula, a, b and c are the numbers in the quadratic equation in standard form, ax² + bx + c.

solving 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.