If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then form an equation whose roots are:
$\frac{α}{β}, \frac{β}{α}$

Open in App

Solution

Since $α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then,

$α + β = - \frac{b}{a}$

And,

$α β = \frac{c}{a}$

If the roots of any equation is $\frac{α}{β}$ and $\frac{β}{α}$ , then,

$(x - \frac{α}{β}) (x - \frac{β}{α}) = 0$

$x^{2} - \frac{β}{α} x - \frac{α}{β} x + 1 = 0$

$α β x^{2} - x (α^{2} + β^{2}) + α β = 0$

$α β x^{2} - x (α^{2} + β^{2} + 2 α β - 2 α β) + α β = 0$

$α β x^{2} - x ({(α + β)}^{2} - 2 α β) + α β = 0$

$\frac{c}{a} x^{2} - x ({(\frac{- b}{a})}^{2} - 2 \frac{c}{a}) + \frac{c}{a} = 0$

$\frac{c}{a} x^{2} - x (\frac{b^{2}}{a^{2}} - \frac{2 c}{a}) + \frac{c}{a} = 0$

$\frac{c}{a} x^{2} - x (\frac{b^{2} - 2 a c}{a^{2}}) + \frac{c}{a} = 0$

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

Therefore, the required equation is $a c x^{2} - x (b^{2} - 2 a c) + a c = 0$ .

Suggest Corrections

Similar questions

Q. If

α

and

β

are roots of

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

, then equation whose roots are

α + \frac{1}{β}, β + \frac{1}{α}

, is

If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ such that $β < α < 0,$ then the quadratic equation whose roots are $| α |, | β |$ is given by

Q. If the roots of the equation

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

are

α

and

β

, then the quadratic equation whose roots are

- α

and

- β

Join BYJU'S Learning Program

If α,β are the roots of the equation ax2+bx+c=0, then form an equation whose roots are:αβ,βα

If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then form an equation whose roots are:
$\frac{α}{β}, \frac{β}{α}$