Verify that roots of the quadratic equation (p - q)x² + (q - r)x + (r - p) = 0 are equal when q + r = 2p.

D=(q+r-2p)²=0 when q+r=2p, so the roots are equal.

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/4/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2026: question 21

Set in the CBSE Class 10 board exam of 2026, paper 30/4/1, section B, question 21, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21 · 2 marks · Very short answerQuadratic Equations

Verify that roots of the quadratic equation $(p - q) x^{2} + (q - r) x + (r - p) = 0$ are equal when $q + r = 2 p$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Equal roots means the discriminant $b^{2} - 4 a c$ must be zero.

Hint 2

Expand $(q - r)^{2} - 4 (p - q) (r - p)$ and try to write it as a perfect square involving $q + r$ and $p$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Compare with $a x^{2} + b x + c = 0$ : $a = p - q$ , $b = q - r$ , $c = r - p$ . The discriminant is $D = b^{2} - 4 a c = (q - r)^{2} - 4 (p - q) (r - p)$ .
½ mark
Expand: $D = q^{2} - 2 q r + r^{2} - 4 (p r - p^{2} - q r + pq) = q^{2} + 2 q r + r^{2} - 4 p (q + r) + 4 p^{2} = (q + r - 2 p)^{2}$ .
½ mark
Put $q + r = 2 p$ : $D = (2 p - 2 p)^{2} = 0$ .
½ mark
Since $D = 0$ , the roots of the given equation are real and equal. Hence verified.
½ mark

Answer: $D = (q + r - 2 p)^{2} = 0$ when $q + r = 2 p$ , so the roots are equal.

Where the marks are

Writes $D = (q - r)^{2} - 4 (p - q) (r - p)$ (½ mark)
Simplifies to $(q + r - 2 p)^{2}$ (½ mark)
Substitutes $q + r = 2 p$ to get $D = 0$ (½ mark)
Concludes that the roots are equal (½ mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Nature of Roots and Discriminant

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2026: question 21

Hints

Step-by-step solution

Revise the chapter

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