First slide

Theory of equations

Question

If the sum of the two roots of the equation $4 x^{3} + 16 x^{2} - 9 x - 36 = 0$ is zero, then the roots are

Moderate

A

$1, 2, - 2$
B

$- 2, \frac{2}{3}, - \frac{2}{3}$
C

$- 3, \frac{3}{2}, - \frac{3}{2}$
D

$- 4, \frac{3}{2}, - \frac{3}{2}$

Solution

Given equation $4 x^{3} + 16 x^{2} - 9 x - 36 = 0,$
Putting $x = - 4 \Rightarrow - 4 \times 64 + 256 + 36 - 36 = 0$
Hence $x = - 4$ is a root of the equation
Now reduced equation is $4 x^{2} (x + 4) - 9 (x + 4) = 0$
$\Rightarrow (x + 4) (4 x^{2} - 9) = 0 \Rightarrow x = - 4, x = \pm \frac{3}{2}$
Thus roots are $- 4, - \frac{3}{2}, \frac{3}{2}$

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