Quadratics and repeated roots
Use the discriminant to interpret how a quadratic meets the horizontal axis.
Pure Mathematics 1 (Paper 1) · Mathematics 9709, 2026–2027. This short revision lesson focuses on discriminants and repeated roots; it does not replace the full section 1.1 pathway. Written solutions use self-check criteria, not an official mark scheme. Specialist accuracy review has not been completed.
Before you begin
- Expand a squared bracket and square a negative number.
- Rearrange an equation so that one side is zero before identifying its coefficients.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For 3x² − 8x + 2 = 0, which substitution correctly finds the discriminant?
What you will learn
- Identify a, b and c correctly.
- Use the discriminant to find a parameter.
What the discriminant tells you
For ax²+bx+c=0 with a≠0, the quantity b²−4ac appears under the square root in the quadratic formula. A positive value produces two distinct real roots, zero produces one repeated real root, and a negative value produces no real roots.
A repeated root corresponds to the graph touching the x-axis at its vertex. This provides a geometric interpretation of setting the discriminant equal to zero.
Parameters are coefficients too
If one coefficient contains an unknown parameter, substitute it into b²−4ac just as you would a number. Solve the resulting equation or inequality and then check it against the requested root condition.
Sources for this lesson
Pause and explain
The diagram shows A: y = (x − 2)² − 2, B: y = (x − 2)² and C: y = (x − 2)² + 2. Which graph has a repeated real root, and why?
Worked example
Find k so that x²−10x+k=0 has equal roots.
Show the worked solution
- Identify a=1, b=−10, c=k.
- Set (−10)²−4k=0.
- Solve 100−4k=0; the resulting quadratic is (x−5)².
Answer k=25
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For y = 2x² − 12x + k, find k so that the graph touches the x-axis. Give the touching point and verify your result by factorising.
Give me a hint
Touching means a repeated root. Use Δ = 0, then find x = −b/(2a).
Compare my reasoning
- Identify a = 2, b = −12 and c = k. The discriminant is 144 − 8k.
- Set 144 − 8k = 0, giving k = 18. The repeated root is x = 12/4 = 3.
- With k = 18, y = 2(x − 3)². It is non-negative and equals zero only at x = 3, so the touching point is (3, 0).
k = 18; touching point (3, 0); y = 2(x − 3)².
Look for these in your work
- Show the coefficient substitution and the condition Δ = 0.
- Find the root as well as the parameter.
- Expand the factorisation to check all three coefficients.
Find m so that 3x² + 12x + m = 0 has equal roots. State the repeated root and verify the factorisation.
Give me a hint
Keep the distinction between a coefficient m and a root x.
Compare my reasoning
- The discriminant is 12² − 4 × 3 × m = 144 − 12m.
- Equal roots require 144 − 12m = 0, so m = 12. Then x = −12/6 = −2.
- Substitution gives 3x² + 12x + 12 = 3(x + 2)². Expanding verifies the original coefficients.
m = 12; repeated root x = −2; 3(x + 2)² = 0.
Look for these in your work
- State why the discriminant must be zero.
- Keep the minus sign in −b/(2a).
- Check your parameter and root in the original equation.
Consider (p − 1)x² + 4x + 4 = 0. Find p for a repeated quadratic root. Then explain why p = 1 also gives exactly one real solution but not a repeated quadratic root.
Give me a hint
Check that the coefficient of x² is non-zero before using a quadratic-root argument.
Compare my reasoning
- For a quadratic, p ≠ 1. Its discriminant is 16 − 16(p − 1) = 32 − 16p.
- Set Δ = 0 to obtain p = 2, which satisfies p ≠ 1. The equation becomes (x + 2)² = 0, with repeated root x = −2.
- At p = 1, the x² term disappears. The linear equation 4x + 4 = 0 has the single solution x = −1; it is not a quadratic and has no repeated quadratic root.
p = 2 gives a repeated quadratic root x = −2. At p = 1 the equation is linear, with x = −1.
Look for these in your work
- Separate the quadratic case from the zero-leading-coefficient case.
- Verify the repeated-root case by factorisation.
- Explain why one real solution alone is insufficient evidence of a repeated quadratic root.
Common mistakes
- Square the whole negative coefficient.
- Equal roots require Δ=0, not c=0.
What can you explain now?
How many distinct real roots does 2x²+3x+2=0 have?
Compare with the explanation
None
Its discriminant is 9−16=−7, which is negative.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Without notes, sketch a quadratic with two distinct roots, a repeated root and no real roots. Label the sign of Δ in each case. Then explain why a ≠ 0 must be checked when the leading coefficient contains a parameter.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.