Question

If Alfa and beta are the zeros of the quadratic polynomial f(x)=ax2+bx+c,then evaluate:(i)alfa-beta,(ii)1/alfa-1/beta

Answer 1

α and β are zeros of quadratic polynomial f(x) = ax² + bx + c
∴sum of roots = -coefficient of x/coefficient of x²
⇒ α + β = - b/a --------(1)

product of roots = constant/Coefficient of x²
αβ = c/a -----------(2)

∵ (α - β)² = (α + β)² - 4αβ
∴ (α - β) = ± √{(α + β)² - 4αβ}
Putting equations (1) and (2),
⇒α - β = ± √{(-b/a)² - 4(c/a)}
= ±√{(b² - 4ac)/a²}
= ± √(b² - 4ac)/a

(ii) 1/α - 1/β = (β - α)/αβ
= -(α - β)/αβ
Use values of (α - β) and αβ
= ±√(b² - 4ac)/a.c/a
= ±√(b² - 4ac)/c

Answer 2

α and β are zeros of quadratic polynomial f(x) = ax² + bx + c

sum of roots = -coefficient of x/coefficient of x²
 α + β = - b/a --------(1)

product of roots = constant/Coefficient of x²
αβ = c/a -----------(2)

We know,  (a - b)² = (a + b)² - 4ab
∴ (α - β) = ± √{(α + β)² - 4αβ}
α - β = ± √{(-b/a)² - 4(c/a)}
= ±√{(b² - 4ac)/a²}
= ± √(b² - 4ac)/a

(ii) 1/α - 1/β = (β - α)/αβ
= -(α - β)/αβ

Putting values of (α - β) and αβ
= ±√(b² - 4ac)/a.c/a
= ±√(b² - 4ac)/c