document.write( "Question 415764: Solve for x: √(x+7)=x-5 \n" ); document.write( "

Algebra.Com's Answer #291705 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here's a procedure for solving these kinds of equations:

Isolate a square that has the variable in its radicand. (\"Radicand\" is the name fo the expression within a radical.) Coefficients in front of the isolated square root are OK.
Square both sides of the equation.
If there is still a square root with a variable in its radicand, repeat steps 1 and 2.
At this point you should have an equation that has no square roots with the variable in its radciand. Use appropriate techniques to solve this equation.
Check you answer(s). This is not optional! Whenever you square both sides of an equation, like we have done a least once at step 2, extraneous solutions can be introduced. Extraneous solutions are solutions that fit the squared equation but do not fit the original equation. Extraneous solutions can happen even if no mistakes were made. So even expert mathematicians must check their answers on thse problems. Extraneous solutions, if any, must be rejected.

\n" ); document.write( "Let's see how this works on your equation:
\n" ); document.write( " $\"sqrt%28x%2B7%29+=+x-5\"$
\n" ); document.write( "1) Isolate a square root.
\n" ); document.write( "Your only square root is already isolated,

\n" ); document.write( "2) Square both sides:
\n" ); document.write( " $\"%28sqrt%28x%2B7%29%29%5E2+=+%28x-5%29%5E2\"$
\n" ); document.write( "The left side is easy to square. Since exponents do not distribute squaring the right side is a little harder than it may seem. To square the right side correctly we can use FOIL on (x-7)(x-7) or use the $\"%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2\"$ pattern with the \"a\" being x and the \"b\" being 5. I like to use the patterns:
\n" ); document.write( " $\"x%2B7+=+%28x%29%5E2+-2%28x%29%285%29+%2B+%285%29%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"x%2B7+=+x%5E2+-10x+%2B+25\"$

\n" ); document.write( "3) If there are square roots left...
\n" ); document.write( "There are no square roots left so we can proceed to step 4.

\n" ); document.write( "4) Solve the equation.
\n" ); document.write( "The equation is a quadratic equation. So we want one side to be zero. Subtracting x and 7 from each side we get:
\n" ); document.write( " $\"0+=+x%5E2+-11x+%2B+18\"$
\n" ); document.write( "Now we factor (or use the Quadratic Formula). This factors easily:
\n" ); document.write( "0 = (x-9)(x-2)
\n" ); document.write( "From the Zero Product Property we know that one of the factors must be zero. So:
\n" ); document.write( "x-9 = 0 or x-2 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x = 9 or x = 2

\n" ); document.write( "5) Check your answer(s).
\n" ); document.write( "Always use the original equation to check:
\n" ); document.write( " $\"sqrt%28x%2B7%29+=+x-5\"$
\n" ); document.write( "Checking x = 9:
\n" ); document.write( " $\"sqrt%28%289%29%2B7%29+=+%289%29-5\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"sqrt%2816%29+=+4\"$
\n" ); document.write( "4 = 4 Check!

\n" ); document.write( "Checking x = 2:
\n" ); document.write( " $\"sqrt%28%282%29%2B7%29+=+%282%29-5\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"sqrt%289%29+=+-3\"$
\n" ); document.write( "3 = -3 Check failed!

\n" ); document.write( "So x = 2 is an extraneous solution which we must reject. (Again, this does not mean we made a mistake earlier.) So the only solution to your equation is x = 9. \n" ); document.write( "

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