Final answer:
The value of p for which the quadratic equation px² + (p+1)x + p = 0 has equal roots is determined by setting the discriminant to zero and solving for p. Upon calculation, the value of p is found to be -1/3.
Step-by-step explanation:
If the quadratic equation px² + (p+1)x + p = 0 has equal roots, the discriminant of the equation must be zero. The discriminant (Δ) is given by Δ = b² - 4ac. For the given equation, a = p, b = p+1, and c = p.
Setting the discriminant to zero, we get:
Setting the discriminant to zero, we get:
(p+1)² - 4(p)(p) = 0
This simplifies to:
p² + 2p + 1 - 4p² = 0
-3p² + 2p + 1 = 0
Factoring or using the quadratic formula:
p = (-2 ± √((2)² - 4(-3)(1))) / (2(-3))
p = (-2 ± √(4 + 12)) / (-6)
p = (-2 ± √(16)) / (-6)
p = (-2 ± 4) / (-6)
Since we need a real value for p that makes sense in the context of the equation, we use:
p = (-2 + 4) / (-6)
p = 2 / (-6)
p = -1/3
Thus, the value of p is -1/3 when the equation has equal roots.