> ## Content Index
> Fetch the complete content index at: https://www.esheets.io/llms.txt
> Use this file to discover other available public pages before exploring further.

# Solving quadratic equations graphically
- URL: https://www.esheets.io/solving-quadratic-equations-graphically/
- Published: 2026-05-17T13:02:45.000Z
- Updated: 2026-06-21T17:07:48.000Z
- Author: Richard Linnington 2
- Tags: Maths, Algebra

Quadratic graphs are useful because they let us see the solutions to an equation. When a quadratic equation is equal to 0, the solutions are the points where the curve crosses the x-axis. These are called the roots or x-intercepts.

On this worksheet, use the graphs to estimate the two x-values that solve each equation. Your answers do not need to be perfect, but they should be sensible estimates from the graph. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise solving quadratic equations using graphs with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Reading a quadratic graph.
- Finding where the graph crosses the x-axis.
- Using x-intercepts as solutions.
- Estimating roots from the graph where needed.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Use each graph to estimate the x-values where the curve meets the required y-value. If the equation equals 0, look for where the curve crosses the x-axis. If the equation equals another number, look across from that y-value and read the matching x-values.  
  
Estimates are acceptable and answers may be entered in either order. 

## Topic guide

Solving a quadratic equation using a graph means finding the x-values on a drawn curve that match a specific condition given by the equation.

When you have a plotted quadratic graph, you can use it to solve related equations without doing any complex algebra:

- If the equation equals zero (for example, x² - 5x + 6 = 0), you are looking for the points where the curve crosses the x-axis.
- If the equation equals a different number (for example, x² - 2x - 1 = 1), you need to find where the curve has a y-value of 1\. You look across from 1 on the y-axis to the curve, and then read the matching x-values.

### Key method

- Look at the number on the right-hand side of your equation.
- Find that number on the y-axis of your graph.
- Draw a horizontal line across from that y-value until you hit the curve. (If the equation equals zero, this line is just the x-axis).
- From the point, or points, where you hit the curve, read directly down or up to the x-axis to find your solutions.

### Worked example

Suppose you have the graph of y = x² + 2x - 3 and you are asked to solve the equation x² + 2x - 3 = 5.

- First, locate 5 on the y-axis.
- Draw a horizontal line straight across from y = 5.
- Find where this line crosses the U-shaped curve. It should cross in two places.
- Read the x-coordinates for both of those crossing points. You might find they are roughly x = 2.4 and x = -4.4\. These are your two estimates.

### Common mistakes and useful tips

- **Reading the wrong axis:** Always make sure you give the x-values for your final answers, not the y-values.
- **Forgetting the second solution:** A U-shaped curve usually crosses a horizontal line twice, meaning there are two valid answers. Make sure you find both of them.
- **Always looking at the x-axis:** It's tempting to always look for the x-intercepts, but you only do this when the equation is equal to 0.
- **Being too precise:** Remember that reading from a graph only gives an estimate. Do not try to guess multiple decimal places; one decimal place is usually enough depending on the grid scale.
- **Answer order:** When you have two solutions, it does not matter which order you write them in.