Q) 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.
PYQ: 32 (b) – CBSE 2025 – Code 30 – Series 5 – Set 1
Ans:
Step 1: Finding value of p:
We know that in a standard quadratic equation, a x2 + b x + c = 0 will have real and equal roots when its discriminant is 0.
i.e. D = b2 – 4 a c = 0
Comparing the given equation (p + 4) x2 – (p + 1) x + 1 = 0 with standard equation, we get:
a = p + 4, b = – (p + 1) and c = 1
For this quadratic equation to have real and equal roots, its discriminant should be zero.
∴ (- (p + 1))2 – 4 (p + 4) (1) = 0
∴ (p + 1)2 – 4 (p + 4) = 0
∴ p2 + 2 p + 1 – 4 p – 16 = 0
∴ p2 – 2 p – 15 = 0
By mid segment splitting, we write this equation as:
∴ p2 – 5 p + 3 p – 15 = 0
∴ p (p – 5) + 3 (p – 5) = 0
∴ (p – 5) (p + 3) = 0
∴ p = 5 and p = – 3
Therefore for p = 5 and p = – 3, the given quadratic equation will have real and equal roots.
Step 2: Finding value of roots:
First, let’s start by substituting the value of p = 5 in the given quadratic equation:
(p + 4) x2 – (p + 1) x + 1 = 0
((5) + 4) x2 – ((5) + 1) x + 1 = 0
9 x2 – 6 x + 1 = 0
By comparing with the standard quadratic equation a x2 + b x + c = 0, we get:
a = 9, b = – 6, c = 1
The value of root is given by: x = [- b ± √ (b 2 – 4 a c)] / 2 a
Since we already know that for p = 5, discriminant D is zero
∴ (b 2 – 4 a c) = 0
∴ x = (- b ± 0) / 2 a
∴ x = (- b + 0) / 2 a and x = (- b – 0) / 2 a
∴ x = – b / 2 a and x = – b / 2 a (both roots are equal)
∵ – b / 2 a = – (- 6) / 2 (9) = 6 / 18 = 1 / 3
∴ x = 1/3 and x = 1/3
Therefore, for p = 5, the roots are 1/3 and 1/3
Next, let’s substitute the value of p = – 3 in the given quadratic equation:
(p + 4) x2 – (p + 1) x + 1 = 0
((- 3) + 4) x2 – ((- 3) + 1) x + 1 = 0
x2 + 2 x + 1 = 0
(x + 1) 2 = 0
x = – 1 and x = – 1
Therefore, for p = – 3, the roots are – 1 and – 1.
Check: Since roots are real and equal for p = 5 as well as p = – 3. Hence our both answers are correct.
Please press the “Heart” button if you like the solution.