For what value of $a$ is the difference between the roots of the equation $(a - 2) x^{2} - (a - 4) x - 2 = 0$ equal to $3$ ?

Open in App

Solution

$(a - 2) x^{2} - (a - 4) x - 2 = 0$
$α + β = \frac{a - 4}{a - 2}$
$α β = \frac{- 2}{a - 2}$
$(α - β)^{2} = α^{2} + β^{2} - 2 α β$
$= (α + β)^{2} - 4 α β$
$= {(\frac{a - 4}{a - 2})}^{2} + \frac{8}{a - 2}$
$\frac{(a^{2} - 8 a + 16) (a - 2) + 8 a - 16}{(a - 2)^{3}}$
$= \frac{a^{3} - 2 a^{2} - 8 a^{2} + 16 a + 16 a - 32 + 8 a - 16}{(a - 2)^{3}}$
$9 = \frac{a^{3} - 10 a^{2} + 40 a - 48}{(a - 2)^{3}}$
$9 a^{3} - 44 a^{2} + 108 a - 72 = a^{3} - 10 x^{2} + 40 x - 48$ .
$8 a^{3} - 44 a^{2} + 68 a - 24 = 0$
$2 a^{3} - 11 a^{2} + 17 a - 6 = 0$
$a = \frac{1}{2}, 2, 3$

Suggest Corrections

Similar questions

a

(a - 2) x^{2} - (a - 4) x - 2 = 0

3

a

2 x^{2} - (a + 1) x + (a - 1) = 0

-3 is a root of the quadratic equation 2x² +px - 15 = 0. For what value of q , the equation p(x² + x ) + q= 0 has equal roots?

2 x^{2} - (a + 1) x + (a - 1) = 0

2 x^{2} + (a + 1) x + (a - 1) = 0

a

Join BYJU'S Learning Program