Question

alpha and beta are the zeroes of the quadratic polynomial ax2 +bx+c then evalute the following :alpha-beta

Answer 1

ax²+bx+c=0
zeroes are α and β
sum of roots = α+β = -b/a
products of roots = ab = c/a
(α+β)² = α²+β² + 2αβ
α²+β² = b²-2ac/a²        ...........(1)

(α-β)² = α²+β² - 2αβ = b² - 4ac / a²            from (1)

α-β  = √ b² -4ac / a²

hope this helps.

Answer 2

Alpha and beta are the zeroes of the quadratic polynomial ax²+bx+c

alpha+beta = -b/a

(alpha)(beta) = c/a

(alpha+beta)² = alpha²+beta²+2(alpha)(beta)

alpha²+beta² = (-b/a)²-2(c/a)

→ = (-b/a)²-2c/a

→ = (-b²-2ac)/a²

(alpha-beta)² = alpha²+beta²-2(alpha)(beta)

→ = {(-b²-2ac)/a²}-2(c/a)

→ = {-b²-2ac}/a² - 2c/a

→ = (-b²-4ac)/a²

(alpha-beta) = √({-b²-4ac}/a²)

= ±√-b²-4ac/a

Hope it helps.....