Question 97624
Given:
.
{{{sqrt(x - 8) = x - 10}}}
.
Get rid of the radical on the left side by squaring both sides of the equation. On the left
side after squaring you just have x - 8 and on the right side you square the quantity (x - 10).
.
After squaring is completed the equation is:
.
{{{x - 8 = x^2 - 20x + 100}}}
.
Let's get the equation in a more conventional form by merely switching (or transposing) 
sides ... swapping the left and right sides to get:
.
{{{x^2 - 20x + 100 = x - 8}}}
.
Get rid of the x on the right side by subtracting x from both sides. The result is:
.
{{{x^2 - 21x + 100 = -8}}}
.
Now get rid of the -8 on the right side by adding 8 to both sides. When you do that the
equation reduces to:
.
{{{x^2 - 21x + 108 = 0}}}
.
This is quadratic equation in a conventional form. If you play with it a little, you will 
find that it factors to become:
.
{{{(x-12)(x - 9) = 0}}}
.
The left side of this equation will be zero (the same as the right side) if either of the
factors is equal to zero.
.
So possible answers come from setting each of the factors equal to zero to get:
.
{{{x - 12 = 0}}}
.
which is solved by adding 12 to both sides to get:
.
{{{x = 12}}}
.
and another solution is obtained from setting:
.
{{{x - 9 = 0}}}
.
which can be solved by adding 9 to both sides to get:
.
{{{x = 9}}}
.
So we have two possible answers ... x = 12 and x = 9.
.
Check both answers by returning to the equation that you were given and substitute
(one at a time) +12 for x and + 9 for x to see if they both work.
.
If you substitute +12 for x in the equation you were given you get:
.
{{{sqrt(12-8) = 12 - 10}}}
.
which reduces to:
.
{{{sqrt(4) = 12 -10}}}
.
and this furthermore reduces to:
.
{{{2 = 2}}}
.
So this answer checks.
.
Now try letting x = 9. Substitute 9 for x in the given equation and you get:
.
{{{sqrt(9-8) = 9 - 10}}}
.
This simplifies to:
.
{{{sqrt(1) = -1}}}
.
but the square root of 1 is +1 and this does not equal -1. So the answer x = 9 does not
work out in the equation. Discard this answer. The only answer that satisfies the original
equation is x = 12.
.
Hope this helps you to understand the problem a little better.