Question

Solve: √(2x ^ 2 - 9x + 25) - √(2x ^ 2 - 9x + 13) = 2​

Answer 1

$\large\underline{\sf{Solution-}}$

Given equation is

$\rm :\longmapsto\: \sqrt{ {2x}^{2} - 9x + 25 } - \sqrt{ {2x}^{2} - 9x + 13 } = 2$

To solve this, Let assume that

$\red{\rm :\longmapsto\:{2x}^{2} - 9x = y}$

So, given equation can be rewritten as

$\rm :\longmapsto\: \sqrt{y + 25} - \sqrt{y + 13} = 2$

$\rm :\longmapsto\: \sqrt{y + 25} = 2 + \sqrt{y + 13}$

On squaring both sides, we get

$\rm :\longmapsto\:y + 25 = {(2 + \sqrt{y + 13} )}^{2}$

$\rm :\longmapsto\:y + 25 = 4 + (y + 13) + 4 \sqrt{y + 13}$

$\rm :\longmapsto\:y + 25 = 4 + y + 13 + 4 \sqrt{y + 13}$

$\rm :\longmapsto\: 25 = 17 + 4 \sqrt{y + 13}$

$\rm :\longmapsto\: 25 - 17 = 4 \sqrt{y + 13}$

$\rm :\longmapsto\: 8= 4 \sqrt{y + 13}$

$\rm :\longmapsto\: 2= \sqrt{y + 13}$

On squaring both sides, we get

$\rm :\longmapsto\:4 = y + 13$

$\rm :\longmapsto \: y = 4 - 13$

$\rm :\longmapsto \: y = - \: 9$

On substituting the value of y, we get

$\rm :\longmapsto\: {2x}^{2} - 9x = - 9$

$\rm :\longmapsto\: {2x}^{2} - 9x + 9 = 0$

$\rm :\longmapsto\: {2x}^{2} - 6x - 3x + 9 = 0$

$\rm :\longmapsto\:2x(x - 3) - 3(x - 3) = 0$

$\rm :\longmapsto\:(2x - 3)(x - 3)= 0$

$\bf\implies \:x = 3 \: \: \: \: or \: \: \: \: x = \dfrac{3}{2}$

Verification :-

Case : - 1

$\red{\rm :\longmapsto\:When \: x = 3}$

On substituting, this value in

$\rm :\longmapsto\: \sqrt{ {2x}^{2} - 9x + 25 } - \sqrt{ {2x}^{2} - 9x + 13 } = 2$

$\rm :\longmapsto\: \sqrt{ {2(3)}^{2} - 9(3) + 25 } - \sqrt{ {2(3)}^{2} - 9(3) + 13 } = 2$

$\rm :\longmapsto\: \sqrt{18 - 27 + 25} - \sqrt{18 - 27 + 13} = 2$

$\rm :\longmapsto\: \sqrt{16} - \sqrt{4} = 2$

$\rm :\longmapsto\:4 - 2 = 2$

$\rm :\longmapsto\:2 = 2$

Hence, Verified

Case :- 2

$\red{\rm :\longmapsto\:When \: x \: = \: \dfrac{3}{2}}$

On substituting the value,

$\rm :\longmapsto\: \sqrt{ {2x}^{2} - 9x + 25 } - \sqrt{ {2x}^{2} - 9x + 13 } = 2$

$\rm :\longmapsto\: \sqrt{2 \times \dfrac{9}{4} - 9 \times \dfrac{3}{2} + 25 } - \sqrt{2 \times \dfrac{9}{4} - 9 \times \dfrac{3}{2} + 1 3} = 2$

$\rm :\longmapsto\: \sqrt{\dfrac{9}{2} - \dfrac{27}{2} + 25} - \sqrt{\dfrac{9}{2} - \dfrac{27}{2} + 13} = 2$

$\rm :\longmapsto\: \sqrt{- \dfrac{18}{2} + 25} - \sqrt{ - \dfrac{18}{2} + 13} = 2$

$\rm :\longmapsto\: \sqrt{ - 9 + 25} - \sqrt{ - 9 + 13} = 2$

$\rm :\longmapsto\: \sqrt{16} - \sqrt{4} = 2$

$\rm :\longmapsto\:4 - 2 = 2$

$\rm :\longmapsto\:2= 2$

Hence, Verified