Find the value(s) of p for which the quadratic equation given as (p+4)x²-(p+1)x+1=0 has real and equal roots. Also, find the roots of the equation(s) so obtained.

p = 5 (roots 1/3, 1/3) and p = −3 (roots −1, −1)

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 32(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(b) · 5 marks · Long answerQuadratic Equations

Find the value(s) of p for which the quadratic equation given as $(p + 4) x^{2} - (p + 1) x + 1 = 0$ has real and equal roots. Also, find the roots of the equation(s) so obtained.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/5/1.

Hints

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

Hint 1

Equal real roots means the discriminant is zero.

Hint 2

Identify a, b, c in terms of p and form a quadratic in p.

Hint 3

Put each value of p back into the equation and solve it.

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.

For real and equal roots, the discriminant $D = b^{2} - 4 a c = 0$ .
½ mark
Here $a = p + 4$ , $b = - (p + 1)$ , $c = 1$ , so $[- (p + 1)]^{2} - 4 (p + 4) (1) = 0$ .
½ mark
$p^{2} + 2 p + 1 - 4 p - 16 = 0 \Rightarrow p^{2} - 2 p - 15 = 0 \Rightarrow (p - 5) (p + 3) = 0$
1 mark
$∴ p = 5$ or $p = - 3$ (both make $p + 4 \neq = 0$ ).
1 mark
For $p = 5$ : $9 x^{2} - 6 x + 1 = 0 \Rightarrow (3 x - 1)^{2} = 0 \Rightarrow x = \frac{1}{3}, \frac{1}{3}$
1 mark
For $p = - 3$ : $x^{2} + 2 x + 1 = 0 \Rightarrow (x + 1)^{2} = 0 \Rightarrow x = - 1, - 1$
1 mark

Answer: p = 5 (roots $\frac{1}{3}, \frac{1}{3}$ ) and p = −3 (roots −1, −1)

Where the marks are

States the condition D = 0 (½ mark)
Substitutes correctly into $b^{2} - 4 a c$ (½ mark)
Forms and factorises $p^{2} - 2 p - 15 = 0$ (1 mark)
Gets p = 5, −3 (1 mark)
Roots $\frac{1}{3}, \frac{1}{3}$ for p = 5 (1 mark)
Roots −1, −1 for p = −3 (1 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 paper2025 board paper, set 30/5/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2025: question 32(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Quadratic Equations