Question

How do you solve `sqrt(9x-8)=x`?

Answer 1

`{(x = 8), (x=1) :}`

Explanation:

Right from the start, you know that your solution(s) must satisfy two conditions

• `x>=0`
• `9x-8>=0 implies x>=8/9`

Overall, you need your solution(s) to satisfy `x>=8/9`.

Start by squaring both sides of the equation to get rid of the radical term

`(sqrt(9x-8))^2 = x^2`

`9x-8 = x^2`

This is equivalent to

`x^2 - 9x + 8 = 0`

The two solutions to this quadratic can be found using the quadratic formula

`x_(1,2) = (-(-9) +- sqrt( (-9)^2 - 4 * 1 * 8))/(2 * 1)`

`x_(1,2) = (9 +- sqrt(49))/2`

`x_(1,2) = (9 +- 7)/2 = {(x_1 = (9 + 7)/2 = 8), (x_2 = (9-7)/2 = 1) :}`

Since both `x_1` and `x_2` satisfy the condition `x>=8/9`, your original equation will have two solutions

`{(x_1 = color(green)(8)), (x_2 = color(green)(1)):}`

Do a quick check to make sure that the calculations are correct

`sqrt(9 * 8 - 8) = 8`

`sqrt(64) = 8 <=> 8 = 8 color(green)(sqrt())`

and

`sqrt(9 * 1 - 8) = 1`

`sqrt(1) = 1 <=> 1 = 1 color(green)(sqrt())`