If α,β are the roots of x2+ax+b=0. Then prove that βα is a root of the equation bx2+(2b−a2)x+b=0.

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If $α, β$ are the roots of $x^{2} + a x + b = 0$ . Then prove that $\frac{α}{β}$ is a root of the equation $b x^{2} + (2 b - a^{2}) x + b = 0$ .

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Olivia Discussed

α, β

are the roots of

x^{2} + a x + b = 0

. Then prove that

\frac{α}{β}

is a root of the equation

b x^{2} + (2 b - a^{2}) x + b = 0

14 mins ago

Discuss this question LIVE

14 mins ago

Given

α, β

are roots of the equation

x^{2} + a x + b = 0

Then,

α + β = - a, α β = b

.......(1).

Now, we to prove

\frac{α}{β}

to be the root of the equation

b x^{2} + (2 b - a^{2}) x + b = 0

, we have to prove that

b (\frac{α}{β})^{2} + (2 b - a^{2}) \frac{α}{β} + b = 0

b (α^{2} + β^{2}) + (2 b - a^{2}) α β = 0

Now,

b (α^{2} + β^{2}) + (2 b - a^{2}) α β

= b {(α + β)^{2} - 2 α . β} + (2 b - a^{2}) α β

= b {(a)^{2} - 2 b} + (2 b - a^{2}) b

[ Using (1)]

= 0

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Question Text	If $α, β$ are the roots of $x^{2} + a x + b = 0$ . Then prove that $\frac{α}{β}$ is a root of the equation $b x^{2} + (2 b - a^{2}) x + b = 0$ .
Answer Type	Text solution:1
Upvotes	150