Quadratic equations with one solution
Use this page when you want to understand why some quadratics end with one repeated real root rather than two different answers.
Start here if you want the short version before reading the full method.
- A quadratic has one repeated real solution when the discriminant b^2 - 4ac equals 0.
- On a graph, that means the parabola touches the x-axis once at its turning point.
What this topic means and what to look for first.
This case often feels surprising because the solving method still seems to produce 'two' branches, but both branches collapse to the same value.
The repeated-root case is easier to recognise once you connect the algebra to the graph.
One reliable route through the topic.
- 1Put the quadratic in standard form and identify a, b, and c.
- 2Compute the discriminant b^2 - 4ac.
- 3If the discriminant is 0, expect one repeated real root.
- 4Solve the equation and then confirm that both branches lead to the same value.
See the method in action.
x^2 - 6x + 9 = 0
- The discriminant is 36 - 36 = 0.
- That means there is one repeated root.
- The factorisation is (x - 3)^2 = 0, so the repeated root is x = 3.
x^2 + 4x + 4 = 0
- The discriminant is 16 - 16 = 0.
- So there is one repeated root.
- The factorisation is (x + 2)^2 = 0, giving x = -2.
Things that commonly send the method off track.
- Writing the same root twice as if it were two different answers.
- Missing the graph meaning and forgetting that the curve only touches the x-axis once.
- Treating a discriminant of 0 as if it were positive.
Use a short verification pass before moving on.
- Confirm that the discriminant is exactly 0, not just close to 0 after arithmetic slips.
- Check the root by substitution and confirm the equation becomes 0.
- If you sketch the graph, the turning point should lie on the x-axis.
Try a few variations before switching to a calculator or solver tool.
- x^2 - 8x + 16 = 0
- x^2 + 10x + 25 = 0
- 4x^2 - 12x + 9 = 0
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Extra algebra revision resources
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Short answers worth checking.
It means the quadratic has one repeated real root, so both branches of the solution method give the same value.
The parabola touches the x-axis once instead of crossing it at two different points.
Continue with the next closely related topic.
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