Name: Condition for Common Roots of Quadratic Equation
Uploaded: 2019-07-19
Duration: 36 min 41 s
Description: In this video, we will learn the condition for one common root and the condition for both roots common for two given quadratic equations with the help of exa...

Question

If $α, β$ are the roots of the quadratic equation $c x^{2} - 2 b x + 4 a = 0$ , then find the quadratic equation whose roots are $\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β}$ .

Accepted Answer

Given $α, β$ are the roots of the quadratic equation $c x^{2} - 2 b x + 4 a = 0$ .

Then,

Sum of the roots $= α + β = \frac{- (- 2 b)}{c} = \frac{2 b}{c} . . . (i)$

Product of the roots $= α \cdot β = \frac{4 a}{c} . . . (ii)$

We know that the quadratic equation with roots $p, q$ will be of the form $x^{2} - (p + q) x + p \cdot q = 0$ .

The quadratic equation with roots $\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β}$ is $x^{2} - (\frac{1 + α}{1 - α} + \frac{1 + β}{1 - β}) x + \frac{1 + α}{1 - α} \cdot \frac{1 + β}{1 - β} = 0$ .

Hence, to find the required quadratic equation, we need to find the sum and product of the symmetric roots.

Sum of the roots $= \frac{1 + α}{1 - α} + \frac{1 + β}{1 - β} = \frac{(1 + α) (1 - β) + (1 + β) (1 - α)}{(1 - α) (1 - β)}$ .

$= \frac{2 - 2 α β}{1 - (α + β) + α β} = \frac{2 - 2 (\frac{4 a}{c})}{1 - (\frac{2 b}{c}) + \frac{4 a}{c}} = \frac{2 c - 8 a}{4 a - 2 b + c}$ .

Product of the roots $= \frac{1 + α}{1 - α} \cdot \frac{1 + β}{1 - β} = \frac{(1 + α) (1 + β)}{(1 + α) (1 - β)}$ .

$= \frac{1 + (α + β) + α β}{1 - (α + β) + α β} = \frac{1 + \frac{2 b}{c} + \frac{4 a}{c}}{1 - \frac{2 b}{c} + \frac{4 a}{c}} = \frac{4 a + 2 b + c}{4 a - 2 b + c}$ .

$∴$ The quadratic equation with roots $\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β}$ is $x^{2} - (\frac{2 c - 8 a}{4 a - 2 b + c}) x + (\frac{4 a + 2 b + c}{4 a - 2 b + c}) = 0$

$\Rightarrow (4 a - 2 b + c) x^{2} - 2 (c - 4 a) x + (4 a + 2 b + c) = 0$ .

If $α, β$ are the roots of the quadratic equation $c x^{2} - 2 b x + 4 a = 0$ , then find the quadratic equation whose roots are $\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β}$ .

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If α, β are the roots of the quadratic equation cx2-2bx+4a=0, then find the quadratic equation whose roots are 1+α1−α, 1+β1−β.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If $α, β$ are the roots of the quadratic equation $c x^{2} - 2 b x + 4 a = 0$ , then find the quadratic equation whose roots are $\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β}$ .