SOLUTION: |x − 11| = x2 − 11x

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Question 708291: |x − 11| = x2 − 11x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
abs%28x-11%29+=+x%5E2-11x
If this were
abs%28x-11%29+=+35
would it be easier? You probably know that this would be solved using two equations:
x-11+=+35 or x-11+=+-35

Well, the problem we have can be solved in exactly the same way!
x-11+=+x%5E2-11x or x-11+=+-%28x%5E2-11x%29
The trick part is that we need the negative of the whole right side in the second equation. When the right side is just a number, like 35, you can just stick a "-" in front. But with multiple terms, like x%5E2-11x, we need to put parentheses around the right side. With the parentheses, the "-" in front will negate the entire expression, not just the x%5E2.

We now have two quadratic equations to solve. Let's do them one at a time. With the first equation we want one side to be zero. Subtracting the entire left side from both sides we get:
0+=+x%5E2-11x%2B11
This will not factor we we will use the Quadratic Formula:
x+=+%28-%28-11%29+%2B-+sqrt%28%28-11%29%5E2-4%281%29%2811%29%29%29%2F2%281%29
Simplifying...
x+=+%28-%28-11%29+%2B-+sqrt%28121-4%281%29%2811%29%29%29%2F2%281%29
x+=+%28-%28-11%29+%2B-+sqrt%28121-44%29%29%2F2%281%29
x+=+%28-%28-11%29+%2B-+sqrt%2877%29%29%2F2%281%29
x+=+%2811+%2B-+sqrt%2877%29%29%2F2
which is short for:
x+=+%2811+%2B+sqrt%2877%29%29%2F2 or x+=+%2811+-+sqrt%2877%29%29%2F2

Now for the second equation we got earlier:
x-11+=+-%28x%5E2-11x%29
Before we get the zero, let's simplify the right side:
x-11+=+-x%5E2%2B11x%29
Adding x%5E2 and subtracting 11x we get:
x%5E2-11x-11+=+0
Again this will not factor so again we use the formula:
x+=+%28-%28-11%29+%2B-+sqrt%28%28-11%29%5E2-4%281%29%28-11%29%29%29%2F2%281%29
Simplifying...
x+=+%28-%28-11%29+%2B-+sqrt%28121-4%281%29%28-11%29%29%29%2F2%281%29
x+=+%28-%28-11%29+%2B-+sqrt%28121%2B44%29%29%2F2%281%29
x+=+%28-%28-11%29+%2B-+sqrt%28165%29%29%2F2%281%29
x+=+%2811+%2B-+sqrt%28165%29%29%2F2
which is short for:
x+=+%2811+%2B+sqrt%28165%29%29%2F2 or x+=+%2811+-+sqrt%28165%29%29%2F2

So there are four solutions to your equation:
x+=+%2811+%2B+sqrt%2877%29%29%2F2 or x+=+%2811+-+sqrt%2877%29%29%2F2
x+=+%2811+%2B+sqrt%28165%29%29%2F2 or x+=+%2811+-+sqrt%28165%29%29%2F2