document.write( "Question 655229: Square root (x+2)+4=x
\n" ); document.write( "Please show how to solve for x \n" ); document.write( "

Algebra.Com's Answer #409185 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 root.
Square both sides. (Note: Squaring the isolated square root should be easy. Squaring the other side can easily be done incorrectly so be careful.)
If there are still any square roots, then repeat steps 1=3. (Note: Squaring both sides of an equation, when done properly, does not necessarily eliminate all the square roots!)
At this point there should be no square roots left. Use appropriate techniques for whatever type of equation you now have.
Check your answer(s). This is not optional! Whenever you square both sides of an equation, like we have done at least once to get this point, then what are called \"extraneous solutions\" may occur. Extraneous solutions are solutions that fit the squared equation but do not fit the original equation! You must check your answers using the original equation. Reject any and all answers that do not fit the original equation, even if it means rejecting all your answers. (Note: Extraneous solutions that may occur does not mean that it is wrong to square both sides of an equation.)

Let's see this in action:
\n" ); document.write( " $\"sqrt%28x%2B2%29%2B4=x\"$
\n" ); document.write( "1. Isolate a square root.
\n" ); document.write( "There is only one square root so there is no choice to be made. To isolate it we just subtract 4 from each side:
\n" ); document.write( " $\"sqrt%28x%2B2%29=x-4\"$

\n" ); document.write( "2. Square both sides.
\n" ); document.write( " $\"%28sqrt%28x%2B2%29%29%5E2=%28x-4%29%5E2\"$
\n" ); document.write( "Squaring the square root is simple. Squaring the other side correctly requires that we use FOIL on (x-4)(x-4) or use the $\"%28a-b%29%5E2=a%5E2-2ab%2Bb%5E2\"$ pattern. I personally prefer using the pattern:
\n" ); document.write( " $\"x%2B2+=+%28x%29%5E2-2%28x%29%284%29%2B%284%29%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"x%2B2+=+x%5E2-8x%2B16\"$

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

\n" ); document.write( "4. Solve the equation.
\n" ); document.write( "This is a quadratic equation. So we want one side to be zero. Subtractin x and 2 from each side we get:
\n" ); document.write( " $\"0+=+x%5E2-9x%2B14\"$
\n" ); document.write( "Now we factor (or use the Quadratic Formula). This factors fairly easily:
\n" ); document.write( " $\"0+=+%28x-7%29%28x-2%29\"$
\n" ); document.write( "From the Zero Product Property we know that
\n" ); document.write( "x-7 = 0 or x-2 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x = 7 or x = 2

\n" ); document.write( "5. Check (using the original equation):
\n" ); document.write( " $\"sqrt%28x%2B2%29%2B4=x\"$
\n" ); document.write( "Checking x = 7:
\n" ); document.write( " $\"sqrt%28%287%29%2B2%29%2B4=%287%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"sqrt%289%29%2B4=7\"$
\n" ); document.write( " $\"3%2B4=7\"$
\n" ); document.write( " $\"7+=+7\"$ Check!

\n" ); document.write( "Checking x = 2:
\n" ); document.write( " $\"sqrt%28%282%29%2B2%29%2B4=%282%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"sqrt%284%29%2B4=2\"$
\n" ); document.write( " $\"2%2B4=2\"$
\n" ); document.write( " $\"6+=+2\"$ Check fails!! So x = 2 is an extraneous solution which we reject.

\n" ); document.write( "So the only solution to your equation is: x = 7. \n" ); document.write( "

\n" );