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

Question

2119 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-08-10T10:56:30

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

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

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

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

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

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