Solving a Quadratic Equation Using a Graph
(KS4, Year 10)

The Lesson

A quadratic equation can be solved by drawing the equation on a graph and seeing where the curve crosses the x-axis. Imagine you wanted to solve the quadratic equation x2 − 3x + 2. Plot y = x2 − 3x + 2 on a graph and read off where the curve crosses the x-axis.

We can see that x = 1 and x = 2 solve the quadratic equation.

How to Solve Quadratic Equations Using a Graph

Solving a quadratic equation using a graph is easy.

Question

Solve the quadratic equation shown below using a graph.

Step-by-Step:

1

Draw the quadratic equation on a pair of axes.

2

Find the x-coordinate of the first point where the curve crosses the x-axis. The curve crosses the x-axis at x = −1, which is a solution to the quadratic equation.

3

Find the x-coordinate of the second point where the curve crosses the x-axis. The curve crosses the x-axis at x = 3, which is also a solution to the quadratic equation.

Answer:

We have solved the quadratic equation: x = −1, x = 3.

Lesson Slides

Sometimes quadratic equations have repeated roots: the same value of x solves the quadratic equation twice. The slider below shows another real example of how to solve a quadratic equation using a graph. Open the slider in a new tab

3 Cases of Roots on a Graph

There are 3 possible cases for the roots of a quadratic equation.
  • 2 real, distinct roots. Occurs when the curve crosses the x-axis in two places.

  • 1 repeated root. Occurs when the curve touches the x-axis at one point.

  • 2 complex roots. Occurs when the curve does not touch the x-axis at all.

Interactive Widget

You can use this interactive widget to create a graph of your quadratic equation. Use the buttons to change the values of the quadratic equation.
Change the Equation
y
x2
x
y-intercept
Help Us To Improve Mathematics Monster

  • Do you disagree with something on this page?
  • Did you spot a typo?
Please tell us using this form

See Also

What is a graph? What is the x-axis? What is a x-coordinate?