Question

Find the value of 'p', if the following quadratic equations have equal roots:

x² + (p − 3)x + p = 0

Answer 1

x² + (p − 3)x + p = 0

 Here, a = 1, b = (p − 3), c = p

 Since, the roots are equal,

⇒ b² − 4ac = 0

⇒ (p − 3)² − 4(1)(p) = 0

⇒ p² + 9 − 6p − 4p = 0

⇒ p² − 10p + 9 = 0

⇒ p² − 9p − p + 9 = 0

⇒ p(p − 9) − 1(p − 9) = 0

⇒ (p − 9) (p − 1) = 0

⇒ p − 9 = 0 or p − 1 = 0

⇒ p = 9 or p = 1 

Answer 2

x² + (p − 3)x + p = 0

has equal roots, we use the condition for equal roots in a quadratic equation. For a quadratic equation ax² + bx + c = 0, the roots are equal if the discriminant (Δ) is zero:

Δ = b² − 4ac = 0

• a = 1
• b = p − 3
• c = p

Substitute into the discriminant:

Δ = (p − 3)² − 4(1)(p) = 0

(p − 3)² − 4p = 0

p² − 6p + 9 − 4p = 0

p² − 10p + 9 = 0

Now solve this quadratic equation for p.

The values of p that satisfy the condition for equal roots are:

p = 1 and p = 9