Name: Transformation of Roots of a Quadratic Equation
Uploaded: 2019-07-19
Duration: 11 min 45 s
Description: Here an equation and its roots are given, and we have to find new quadratic equations with new roots, which are simple multiples or factors of the original r...

Question

Solve the equation

x^{4} - 3 x^{2} - 6 x - 2 = 0

.

Accepted Answer

Given equation is $x^{4} - 3 x^{2} - 6 x - 2 = 0$ .

Suppose $x^{4} - 3 x^{2} - 6 x - 2 = (x^{2} + p x + q) (x^{2} - p x + q^{'})$

$\Rightarrow x^{4} - 3 x^{2} - 6 x - 2 = x^{4} + (q + q^{'} - p^{2}) x^{2} + (p q^{'} - p q) x + q q^{'}$

Comparing coefficients

$q + q^{'} - p^{2} = - 3$ $. . . (1)$

$\Rightarrow (q + q^{'}) = (p^{2} - 3)$

$p q^{'} - p q = - 6$ $. . . (2)$

$\Rightarrow (q^{'} - q) = \frac{- 6}{p}$

$q q^{'} = - 2$ $. . . (3)$

$∵$ ${(q - q^{'})}^{2} = {(q + q^{'})}^{2} - 4 q q^{'}$

Substitute the values of $(1), (2)$ and $(3)$ we get,

$\frac{36}{p^{2}} = {(p^{2} - 3)}^{2} + 8$

$\Rightarrow \frac{36}{p^{2}} = p^{4} - 6 p^{2} + 17$

$\Rightarrow p^{6} - 6 p^{4} + 17 p^{2} - 36 = 0$

$\Rightarrow t^{3} - 6 t^{2} + 17 t - 36 = 0$ , where $t = p^{2}$ .

By trial, we see that $t = 4$ satisfies above equation.

$∴$ $p^{2} = 4$

$∴$ $p = 2$ or $- 2$

Choosing $p = 2$ we get,

$q + q^{'} = 1, q^{'} - q = - 3$ from $(2)$

$∴$ $q^{'} = - 1$ and $q = 2$

Now,

$x^{4} - 3 x^{2} - 6 x - 2 = (x^{2} + p x + q) (x^{2} - p x + q^{'})$

$\Rightarrow x^{4} - 3 x^{2} - 6 x - 2 \equiv (x^{2} + 2 x + 2) (x^{2} - 2 x - 1) = 0$

$∴$ $x^{2} + 2 x + 2 = 0$ or $x^{2} - 2 x - 1 = 0$

$∴$ $x = \frac{- 2 \pm \sqrt{- 4}}{2}, \frac{2 \pm \sqrt{8}}{2}$

$\Rightarrow x = (- 1 \pm \sqrt{- 1}), 1 \pm \sqrt{2}$

Again, if $p = - 2$ , then

$q + q^{'} = 1, q^{'} - q = 3$

Hence, $q^{'} = 2, q = - 1$

Therefore,

$x^{4} - 3 x^{2} - 6 x - 2 = (x^{2} - 2 x - 1) (x^{2} + 2 x + 2)$

Again, we are getting the same equations, hence same solution.

Solve the equation $x^{4} - 3 x^{2} - 6 x - 2 = 0$ .

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Solve the equation x4-3x2-6x-2=0.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Solve the equation $x^{4} - 3 x^{2} - 6 x - 2 = 0$ .