If α and β are the roots of the equation ax² + bx + c =0, express the roots of the equation a³x² - ab²x + b²c = 0 in terms of α and β
Answers
EXPLANATION.
α and β are the roots of the equation.
⇒ ax² + bx + c = 0.
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
Roots of the equation.
⇒ a³x² - ab²x + b²c = 0.
As we know that,
Let, γ and δ are the roots of the equation.
⇒ a³x² - ab²x + b²c = 0.
Sum of the zeroes of the quadratic polynomial.
⇒ γ + δ = - b/a.
⇒ γ + δ = - (-ab²)/a³ = ab²/a³.
⇒ γ + δ = (b²/a²) = (-b/a)² = (α + β)².
Products of the zeroes of the quadratic polynomial.
⇒ γδ = c/a.
⇒ γδ = (b²c/a³) = (b²/a²)(c/a).
⇒ γδ = (b/a)²(c/a) = (α + β)².(αβ).
⇒ (γ - δ)² = (γ + δ)² - 4γδ.
⇒ (γ - δ)² = [(α + β)²]² - 4(αβ)(α + β)².
⇒ (γ - δ)² = (α + β)⁴ - 4(αβ)(α + β)².
⇒ (γ - δ) = (α² - β²). - - - - - (1).
⇒ (γ + δ) = (α + β)².
⇒ (γ + δ) = α² + β² + 2αβ. - - - - - (2).
Adding equation (1) and (2), we get.
⇒ 2γ = 2α² + 2αβ.
⇒ γ = α² + αβ.
Pu the values of γ = α² + αβ in equation (1), we get.
⇒ (γ - δ) = (α² - β²). - - - - - (1).
⇒ (α² + αβ) - δ = (α² - β²).
⇒ (α² + αβ) - (α² - β²) = δ.
⇒ α² + αβ - α² + β² = δ.
⇒ β² + αβ = δ.
Values of γ = α² + αβ. and β² + αβ = δ.
Step-by-step explanation:
Sol. α + β = -b/a, αβ = c/a
Roots if the equation a3 x3 + abcx + c3 = 0 are
x = -abc ± √(abc)2 – 4a3 c3/2a3
= (-b/a) (c/a) ± √(b/a)2 (c/a)2 – 4(c/a)3 / 2
(α + β) (α β) ± √( α + β)2 (αβ)2 – 4(αβ)3 / 2
= (α β) ((α + β) ± √( α – β)2 / 2
= (α β) ((α + β) ± (α – β)/2 = α2 β, α β2
Let γ and δ be the required roots. Then
γ = α2 β and δ = α β2.
ALTERNATE SOLUTION :
ax2 + bx + c = 0 has roots α and β. (given)
⇒ α + β = -b/a and α β = c/a
Now, a3 x2 + abcx + c3 = 0
Divides the equation by c2, we get
a3/c2 x2 + abcx/c2 + c3/c2 = 0, a(ax/c)2 + b(ax/c) + c = 0
⇒ ax/c = α, β are the roots
⇒ x = c/a α, c/a β are the roots
⇒ x = α β α , α β β are the roots
⇒ x α2 β, α β2 are the roots
ALTERNATE SOLUTION :
Divide the equation by a3, we get
x2 + b/a. c/a. x + (c/a)3 = 0
⇒ x2 – (α + β) (α β)x + (α β)3 = 0
⇒ x2 – α2 βx - α β2 x + (α β)3 = 0
⇒ x (x – α2 β) - α β2 (x – α2 β) = 0
⇒ (x – α2 β) (x - α β2) = 0
⇒ x = α2 β, α β2 which is the required answer