document.write( "Question 351639: Please help me solve for x: \"+sqrt%28+x%2B5%29+-+sqrt%28+x%2B3+%29=+2+\" \n" ); document.write( "
Algebra.Com's Answer #251561 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
To solve equations where the variable is inside a square root:
  1. Isoalte a square root.
  2. Square both sides of the equation.
  3. If there are still any square roots, repeat steps #1 and #2.
  4. At this point the equation should have no square roots remaining. Use appropriate techniques the this equation.
  5. Check your answer(s)! This is important, not just a good idea. Whenever you square both sides of an equation, which has been done at least once to this far, extraneous solutions may have been introduced even if no errors have been made!. Extraneous solution are solutions which fit the squared equation but do not fit the original equation! So we must check any answers and make sure they actually fit the original equation. Any extraneous solutions (which do not fit the original equation) must be rejected, even if it means that no solutions remain!

\n" ); document.write( "Let's see how this works on your equation.
\n" ); document.write( "\"+sqrt%28+x%2B5%29+-+sqrt%28+x%2B3+%29=+2+\"
\n" ); document.write( "1) Isolate a square root. By adding the second square root to each side, the first square root will be isolated on the left side:
\n" ); document.write( "\"+sqrt%28+x%2B5%29+=+sqrt%28+x%2B3+%29%2B+2+\"
\n" ); document.write( "2) Square both sides.
\n" ); document.write( "\"+%28sqrt%28+x%2B5%29%29%5E2+=+%28sqrt%28+x%2B3+%29%2B+2%29%5E2+\"
\n" ); document.write( "With only the square root, the left side will be easy to square. But the right side, with two terms (the square root and the 2), will require either FOIL or the pattern \"%28a%2Bb%29%5E2+=+a%5E2+%2B+2ab+%2B+b%5E2\" to square it properly:
\n" ); document.write( "\"x%2B5+=+%28sqrt%28+x%2B3+%29%29%5E2+%2B+2%2Asqrt%28x%2B3%29%2A2+%2B+2%5E2+\"
\n" ); document.write( "Simplifying we get:
\n" ); document.write( "\"x%2B5+=+x%2B3+%2B+4%2Asqrt%28x%2B3%29+%2B+4\"
\n" ); document.write( "\"x%2B5+=+x%2B7+%2B+4%2Asqrt%28x%2B3%29\"
\n" ); document.write( "3) We still have a square root. So we must repeat steps #1 and #2 until they are all gone. Isolating the remaining square root (by subtracting x and 7 from each side):
\n" ); document.write( "\"-2+=+4sqrt%28x%2B3%29\"
\n" ); document.write( "At this point, if you're thinking about what you see, you will notice something important. You have an equation that says 4 times a square root is equal to -2. But all square roots are positive and 4 times a positive number cannot be equal to -2! This means that there is no solution for this equation. But we will continue as if we didn't notice this.
\n" ); document.write( "The square root is isolated enough. (You could divide both sides by 4 but it isn't necessary. So we can proceed to squaring both sides:
\n" ); document.write( "\"%28-2%29%5E2+=+%284sqrt%28x%2B3%29%29%5E2\"
\n" ); document.write( "\"4+=+16%2A%28x%2B3%29\"
\n" ); document.write( "We finally have an equation without any square roots. So we can proceed with solving it. We'll start by simplifying:
\n" ); document.write( "\"4+=+16x%2B48%29\"
\n" ); document.write( "Subtract 48:
\n" ); document.write( "\"-44+=+16x\"
\n" ); document.write( "Divide by 16:
\n" ); document.write( "\"-44%2F16+=+x\"
\n" ); document.write( "Reducing:
\n" ); document.write( "\"-11%2F4+=+x\"
\n" ); document.write( "4) Check you answer in the original equation:
\n" ); document.write( "\"+sqrt%28+x%2B5%29+-+sqrt%28+x%2B3+%29=+2+\"
\n" ); document.write( "Checking x = -11/4:
\n" ); document.write( "\"+sqrt%28+%28-11%2F4%29%2B5%29+-+sqrt%28+%28-11%2F4%29%2B3+%29=+2+\"
\n" ); document.write( "\"+sqrt%28+%28-11%2F4%29%2B20%2F4%29+-+sqrt%28+%28-11%2F4%29%2B12%2F4+%29=+2+\"
\n" ); document.write( "\"+sqrt%28+9%2F4+%29+-+sqrt%281%2F4%29=+2+\"
\n" ); document.write( "\"3%2F2+-+1%2F2+=+2\"
\n" ); document.write( "\"2%2F2+=+2\"
\n" ); document.write( "\"1+=+2\" Does not check!! So we must reject this solution. And since this was the only \"solution\" we found, the equation has no solutions at all! (This agrees with what we found way back at \"-2+=+4sqrt%28x%2B3%29\".) \n" ); document.write( "
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