Question

Given that p and q are the roots of $2 x^{2} +x - 5=0$, calculate the value of $p^2 - q^2$

Answer 1

If p and q are roots of 2x²+x-5=0

p+q = -b/a = -1/2
pq = c/a = -5/2

(p+q)² = (-1/2)²
⇒ p² + q² + 2pq = 1/4
⇒ p² + q² + 2×(-5/2) = 1/4
⇒ p² + q² - 5 = 1/4
⇒ p² + q² = 5+1/4 
⇒ p² + q² = 21/4

(p-q)² = p² + q² - 2pq = 21/4 - 2×(-5/2) = 21/4 + 5 = 41/4
⇒ p-q = √(41/4)
⇒ p-q = +√(41)/2 or -√(41)/2

p² - q² = (p+q)(p-q) = -1/2 × √(41) / 2 = -√(41) /4  and
p² - q² = (p+q)(p-q) = -1/2 × -√(41) / 2 = √(41) /4

So the product of roots is $\pm \frac{ \sqrt{41} }{4}$