SOLUTION: how do I solve the square root of x+2 -1 outside the root = square root of x-5?

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Question 486950: how do I solve the square root of x+2 -1 outside the root = square root of x-5?
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given to solve:
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sqrt%28x%2B2%29+-+1+=+sqrt%28x-5%29
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Let's begin by squaring both sides. When you square the left side you multiply the entire left side by itself. So the left side is squared by multiplying:
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%28sqrt%28x%2B2%29+-+1%29%2A%28sqrt%28x%2B2%29+-+1%29
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We can do this multiplication by multiplying the first term in the first set of parentheses times both terms in the second set of parentheses as follows:
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Note that the first term on the right side, when squared is just %28x%2B2%29. Therefore, this first multiplication results in:
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%28x%2B2%29+-+sqrt%28x+%2B+2%29 <--- remember this answer
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Not done yet with squaring the left side. Next we have to multiply the second term in the first set of parentheses by both terms in the second set of parentheses. This multiplication is:
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-1%2A%28sqrt%28x%2B2%29+-+1%29+=+-sqrt%28x%2B2%29+%2B+1
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Add this result to the answer we reminded ourselves to remember above:
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Add the two constants (the +2 and the +1) and the answer simplifies to:
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x+%2B+3+-+2sqrt%28x%2B2%29
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This is the left side squared. Now we have to return to the original equation you were given and square the right side. In other words we square sqrt%28x-5%29. And when we square the square root of a quantity the answer is the quantity itself. So:
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%28sqrt%28x-5%29%29%5E2+=+x+-+5
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This is the right side squared. It is equal to what we got for the left side squared. So the "left side squared" equals "right side squared" equation becomes:
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x+%2B+3+-+2sqrt%28x%2B2%29=+x+-5
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We can simplify this by subtracting x from both sides of this equation. Doing this causes the x to disappear on both the left and right sides so we are left with the equation:
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3+-+2sqrt%28x%2B2%29+=+-5
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Next we transfer the 3 to the right side by subtracting 3 from both sides. This subtraction causes the 3 to disappear from the left side and the -5 on the right side becomes -8. So we are left with:
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-2sqrt%28x+%2B+2%29+=+-8
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Then divide both sides by -2 and the equation reduces to:
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sqrt%28x%2B2%29+=+4
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We have just a radical on the left side and we can get rid of it by squaring both the left and right sides. That squaring causes the equation to become:
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x+%2B+2+=+16
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Finally solve for x by subtracting 2 from both sides and the answer is:
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x+=+14
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You can check this answer by returning to the original equation given to you in the problem and substituting 14 for x. You will find that this substitution causes both the left and right sides of the original equation to be equal. This is as follows:
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sqrt%28x%2B2%29+-+1+=+sqrt%28x-5%29
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sqrt%2814%2B2%29+-+1+=+sqrt%2814-5%29
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sqrt%2816%29+-+1+=+sqrt%289%29
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4+-+1+=+3
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3+=+3
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The answer checks. x equals 14.
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Hope this helps you to understand working with radicals a little better.