Solve for x if 4(2x+3)2−(2x+3)−14=0.

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Question

Solve for x if $4 (2 x + 3)^{2} - (2 x + 3) - 14 = 0$ .

Options:

A .

x = - \frac{1}{2}, - \frac{19}{8}

B .

x = \frac{1}{2}, \frac{19}{8}

C .

x = \frac{1}{2}, - \frac{19}{8}

D .

x = - \frac{1}{2}, \frac{19}{8}

Answer: Option A
:
A

Given:   $4 (2 x + 3)^{2} - (2 x + 3) - 14 = 0$
Substitute $(2 x + 3) = y$ , Hence the given equation reduces to
$4 y^{2} - y - 14 = 0$
$\Rightarrow$ $4 y^{2} - 8 y + 7 y - 14 = 0$
$\Rightarrow$ $4 y (y - 2) + 7 (y - 2) = 0$
$\Rightarrow$ $(4 y + 7) (y - 2) = 0$
$\Rightarrow$ $y = \frac{- 7}{4} o r y = 2$
When $y = - \frac{7}{4}$ ,
$(2 x + 3) = - \frac{7}{4}$
$2 x = - \frac{19}{4}$
$\Rightarrow$    $x = - \frac{19}{8}$
When $y = 2$ ,
$(2 x + 3) = 2$
$2 x = - 1$
$\Rightarrow$    $x = - \frac{1}{2}$

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