To solve an equation using two graphs, plot one side of the equation as each graph. Find the points where they meet. The x-coordinates are the solutions because the two sides have the same value there.
The 2027 SEC G3 Mathematics K310 syllabus includes solving quadratic equations graphically. This example uses an existing curve and adds a straight line, so the solutions can be read without plotting a new quadratic.
- Write each side as a separate expression for y.
- Draw or identify both graphs on the same axes.
- Mark every visible intersection.
- Read each intersection’s x-coordinate.
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Decide what to draw on the given axes
Suppose the curve y = x² − 4x is already drawn and the question asks you to solve x² − 4x = 4 − x. Add the line y = 4 − x. You can plot it using (0, 4) and (4, 0). Extend the line across the relevant part of the graph so it can meet both sides of the curve.
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Worked example | Read the two crossing points
The graphs meet at (−1, 5) and (4, 0) in this example. Read the x-coordinates from those points. The solutions are x = −1 and x = 4. At x = −1, both expressions give 5. At x = 4, both give 0.
The first crossing is above the horizontal axis. It still gives a solution, because the two expressions are equal there. Reading only where the curve crosses the x-axis would solve x² − 4x = 0 instead.
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Check the answers without the graph
Rearrange the original equation to x² − 3x − 4 = 0. Factor it as (x − 4)(x + 1) = 0. The roots confirm the two graph readings. On an exam graph, give your readings to a precision supported by the scale.
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Frequently asked questions
When do I draw a horizontal line?
If one side of the equation is a constant, such as 3, use y = 3 as the second graph.
What can a parent ask during revision?
Ask the student to point to a crossing and explain what its x-coordinate and y-coordinate each represent.