Question

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

Answer 1

`a = 7`

Explanation:

Given: `2sqrt (6a + 7) - 2a = 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`

Add "a" to both sides:

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

Square both sides:

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

Subtract `6a+7` from both sides:

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

Factor:

`(a-7)(a+1)=0`

`a = 7 and a = -1`

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

Check:

`2sqrt (6(7) + 7) - 2(7) = 0`
`2sqrt (6(-1) + 7) - 2(-1) = 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.