# How would yous solve sqrt(3x+1)-1=sqrt(8x-1)?

May 15, 2018

$x = 0.129$

#### Explanation:

In the equation $\sqrt{3 x + 1} - 1 = \sqrt{8 x - 1}$ we can only have $3 x + 1 \ge 0$ and $8 x - 1 \ge 0$ i.e. $x \ge - \frac{1}{3}$ and $x \ge \frac{1}{8}$ i.e. $x \ge \frac{1}{8}$ or $x \ge 0.125$.

To solve the equation $\sqrt{3 x + 1} - 1 = \sqrt{8 x - 1}$

squaring each side, we get

$3 x + 1 - 2 \sqrt{3 x + 1} + 1 = 8 x - 1$

now take irrational portion on the left and we get

$- 2 \sqrt{3 x + 1} = 8 x - 1 - 3 x - 1 - 1 = 5 x - 3$

squaring again $4 \left(3 x + 1\right) = 25 {x}^{2} - 30 x + 9$

or $25 {x}^{2} - 30 x + 9 = 12 x + 4$

or $25 {x}^{2} - 42 x + 5 = 0$

or $x = \frac{42 \pm \sqrt{{42}^{2} - 500}}{50} = \frac{42 \pm \sqrt{1264}}{50} = \frac{42 \pm 35.553}{50}$

i.e. $x = 0.129$ or $1.551$

However on checking $1.551$ is not found to be a solution and hence only answer is $x = 0.129$