Question

if alpha and beta are the roots of the ax^2+bx+c=0 then find 1/alpha-1/alpha
please answer it fast​

Answer 1

Question :

If α and β are the roots of ax² + bx + c = 0. Then, find 1/α - 1/β.

Solution:

We know that,

• α + β = -b/c
• αβ = c/a

Now,

$( \alpha - \beta )^{2} = \alpha ^{2} + \beta^{2} - 2 \alpha \beta \: \: \: \: \: \\ \\ ( \alpha - \beta )^{2} =( \alpha + \beta ) ^{2}- 4 \alpha \beta \: \: \: \: \\ \\ ( \alpha - \beta )^{2} = (\frac{ -b}{a}^{2} ) - \frac{4c}{a} \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ ( \alpha - \beta )^{2} = \frac{ b^{2} - 4c}{ {a}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \alpha - \beta = \sqrt{\frac{ b^{2} - 4c}{ {a}^{2} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{\alpha - \beta = \pm \frac{ \sqrt{ b^{2} - 4c}}{ {a}^{2} }}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \:$

Now,

$\dfrac{1}{ \alpha } - \dfrac{1}{ \beta }$

$\implies \dfrac{ \alpha - \beta }{ \alpha \beta }$

$\implies \dfrac{ \pm \: \sqrt{{b}^{2} - 4ac}}{a} \times \dfrac{a}{c}$

$\boxed{ \bf \therefore \: \dfrac{1}{ \alpha } - \frac{1}{ \beta } = \frac{ \pm \: \sqrt{{b}^{2} - 4ac}}{c }}$