# 2sqrt (6a + 7) - 2a = 0? solve the equation

Apr 18, 2017

$a = 7$

#### Explanation:

Given: $2 \sqrt{6 a + 7} - 2 a = 0$

Restrict the domain so that the expression under the radical never becomes negative:

2sqrt (6a + 7) - 2a = 0; a>=-7/6

Divide the equation by 2:

sqrt (6a + 7) - a = 0; a>=-7/6

sqrt (6a + 7) = a; a>=-7/6

Square both sides:

6a + 7 = a^2; a>=-7/6

Subtract $6 a + 7$ from both sides:

0 = a^2-6a - 7; a>=-7/6

Factor:

$\left(a - 7\right) \left(a + 1\right) = 0$

$a = 7 \mathmr{and} a = - 1$

Please notice that I have can discarded the restriction because neither root violates it.

Check:

$2 \sqrt{6 \left(7\right) + 7} - 2 \left(7\right) = 0$
$2 \sqrt{6 \left(- 1\right) + 7} - 2 \left(- 1\right) = 0$

$0 = 0$
$2 = 0$

The negative root must be discarded; it is an erroneous root introduced by squaring.

$a = 7$ is the only correct answer.