Question

if alpha and beta are the zeroes of the polynomial ax^2+bx+c=0 then evaluate alpha - beta

Answer 1

Hey there! 

Refer the Attchment :)

Cheers!

Answer 2

Answer:

$\frac{\sqrt{b^2 - 4ac}}{a}$

Step-by-step explanation:

Given quadratic equation,

$ax^2+bx+c=0$

Since, $\alpha$ and $\beta$ are the roots of the above equation,

$\alpha + \beta = -\frac{b}{a}$ ------(1)

$\alpha.\beta = \frac{c}{a}$ -----(2),

( Equation (1) )² - 4 × Equation (2),

We get,

$(\alpha+\beta)^2- 4 \alpha.\beta = \frac{b^2}{a^2}-\frac{4c}{a}$

$(\alpha)^2 + (\beta)^2 + 2. \alpha.\beta-4 \alpha.\beta = \frac{b^2}{a^2}-\frac{4c}{a}$

$(\alpha)^2 + (\beta)^2 -2 \alpha.\beta = \frac{b^2}{a^2}-\frac{4c}{a}$

$(\alpha - \beta )^2 = \frac{b^2 - 4ac}{a^2}$

$\implies \alpha - \beta = \sqrt{\frac{b^2 - 4ac}{a^2}}=\frac{\sqrt{b^2 - 4ac}}{a}$