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Solving Quadratics Graphically

FoundationHigherAQAEdexcelOCR

Practise solving quadratics graphically with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through solving quadratics using graphs, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. The solutions are where the curve crosses the x-axis.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

Solving a quadratic graphically means reading the solutions from where a curve meets a line. The solutions of \(x^2 - 4 = 0\) are simply where the curve \(y = x^2 - 4\) crosses the \(x\)-axis.

The same idea extends further. To solve \(x^2 - 4 = 3\), draw the horizontal line \(y = 3\) and read where it cuts the curve. The \(x\) values at those intersections are the solutions.

Because you are reading from a drawing, answers are approximate. Questions usually ask for one or two decimal places, and it is worth marking the reading lines on the graph, since method marks are awarded for showing where you read from.

Revision notes

Reading roots from the axis

The roots of \(y = 0\) are where the curve crosses the horizontal axis.

For \(y = x^2 - 9\), the curve crosses at \(x = 3\) and \(x = -3\), so those are the solutions. A quadratic usually has two roots, sometimes one, and occasionally none.

Solving against a horizontal line

To solve an equation equal to a number, draw that horizontal line and read the intersections.

For \(x^2 - 2x = 3\), draw \(y = 3\) across the curve \(y = x^2 - 2x\) and read the two \(x\) values where they meet.

Rearranging to match the drawn curve

If the equation does not match the curve you have, rearrange it so it does, then draw whatever line the remainder describes.

To solve \(x^2 - 3x - 1 = 0\) using the curve \(y = x^2 - 3x\), rewrite it as \(x^2 - 3x = 1\) and draw \(y = 1\).

Key points

  • Roots are where the curve crosses the x-axis.
  • Solve against a number by drawing that horizontal line.
  • Read the x values at the intersections.
  • A quadratic usually has two solutions.
  • Answers from a graph are approximate.
  • Show your reading lines on the diagram.

Worked examples

Example 1

Use the graph of \(y = x^2 - 16\) to solve \(x^2 - 16 = 0\).

Working

\[\text{Read where the curve crosses } y = 0\]the roots are on the x-axis
\[x = 4 \text{ and } x = -4\]state both solutions

Example 2

Explain how to solve \(x^2 + x = 6\) using the curve \(y = x^2 + x\).

Working

\[\text{Draw the line } y = 6\]the equation equals 6, so draw that horizontal line
\[\text{Read the x values where they meet}\]the intersections give the solutions
\[x = 2 \text{ and } x = -3\]state both solutions

Example 3

How many solutions does \(x^2 + 4 = 0\) have graphically?

Working

\[\text{The curve } y = x^2 + 4 \text{ has minimum } 4\]the lowest point is above the x-axis
\[\text{No solutions}\]the curve never crosses y = 0

Common mistakes

  • Giving only one solution.

    A quadratic normally crosses the axis twice, so check both sides of the turning point.

  • Reading the y value instead of the x value.

    The solutions are the x coordinates of the intersections.

  • Not drawing the required line.

    To solve against a number you must draw that horizontal line on the graph.

  • Giving an exact answer from a graph.

    Graphical readings are approximate, so give them to the stated accuracy.

Exam tips

  • Draw and label the line you are solving against.
  • Leave your reading lines visible for the method marks.
  • Check whether the question expects two solutions.
  • State answers to the accuracy the question requires.

Key terms

Root
A solution where the curve crosses the x-axis.
Intersection
A point where two graphs meet.
Approximate
Close to but not exactly the true value.
Rearrange
To rewrite an equation in a more useful form.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.