document.write( "Question 408465: √(1+4√(x))=1+√(x) \n" ); document.write( "

Algebra.Com's Answer #287995 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%281%2B4%2Asqrt%28x%29%29+=+1+%2B+sqrt%28x%29\"$
\n" ); document.write( "Here's a procedure to follow when solving equations of one variable where the variable is found inside one or more square roots:

Isolate a square root that has a variable in its radicand. (The expression inside a radical is called a radicand.)
Square both sides of the equation.
If there is still a square root with the variable in its radicand, then repeat steps 1 and 2.
At this point there should be no square roots with a variable in its radicand. Use appropriate techniques to solve the equation.
Check you solution(s). This is not optional. When you square both sides of an equation, like we have done at least once at step 2, extraneous solutions may be introduced. Extraneous solutions are solutions which fit the squared equation but do not fit the original equation. Extraneous solutions, if any, must be rejected. These extraneous solutions, if any, can happen even if no mistakes have been made! So even Math experts have to check their solutions to equations like this.

Let's see this in action...
\n" ); document.write( "1) Isolate a square root...
\n" ); document.write( "The square root on the left side is already all by itself.

\n" ); document.write( "2) Square both sides.
\n" ); document.write( " $\"%28sqrt%281%2B4%2Asqrt%28x%29%29%29%5E2+=+%281+%2B+sqrt%28x%29%29%5E2\"$
\n" ); document.write( "The left side is simple to square. The right side is not so easy. To square it correctly we should use FOIL or the $\"%28a%2Bb%29%5E2+=+a%5E2+%2B2ab+%2Bb%5E2\"$ pattern. I prefer to use the pattern:
\n" ); document.write( " $\"1%2B4%2Asqrt%28x%29+=+%281%29%5E2+%2B+2%281%29%28sqrt%28x%29%29+%2B+%28sqrt%28x%29%29%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"1%2B4%2Asqrt%28x%29+=+1+%2B+2sqrt%28x%29+%2B+x\"$

\n" ); document.write( "3) There is still a square root (actually two) which has the variable in its radicand. So we repeat steps 1 and 2.

\n" ); document.write( "1) Isolate a square root...
\n" ); document.write( "Subtracting $\"2%2Asqrt%28x%29\"$ we get:
\n" ); document.write( " $\"1%2B2%2Asqrt%28x%29+=+1+%2B+x\"$
\n" ); document.write( "Subtracting 1 from each side we get:
\n" ); document.write( " $\"2%2Asqrt%28x%29+=+x\"$
\n" ); document.write( "The 2 on the left side is not a problem. We will still be able to square both sides and have the square root disappear. But if it bothers you, then just divide both sides by 2:
\n" ); document.write( " $\"sqrt%28x%29+=+x%2F2\"$

\n" ); document.write( "2) Square both sides.
\n" ); document.write( " $\"%28sqrt%28x%29%29%5E2+=+%28x%2F2%29%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"x+=+x%5E2%2F4\"$

\n" ); document.write( "3) There are not more square roots so we can proceed to step 4.

\n" ); document.write( "4) Solve the equation.
\n" ); document.write( "This is a quadratic equation (because of the $\"x%5E2\"$ ) so we want one side to be zero. Subtracting x from each side we get:
\n" ); document.write( " $\"0+=+x%5E2%2F4+-+x\"$
\n" ); document.write( "The equation will be easier to solve without the fraction so we'll multiply both sides by 4:
\n" ); document.write( " $\"0+=+x%5E2+-+4x\"$
\n" ); document.write( "Now we factor (or use the Quadratic Formula). This factors very easily:
\n" ); document.write( "0 = x(x-4)
\n" ); document.write( "From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
\n" ); document.write( "x = 0 or x-4 = 0
\n" ); document.write( "Solving the second equation we get:
\n" ); document.write( "x = 0 or x = 4

\n" ); document.write( "5)Check the solution(s).
\n" ); document.write( "Always use the original equation to check:
\n" ); document.write( " $\"sqrt%281%2B4%2Asqrt%28x%29%29+=+1+%2B+sqrt%28x%29\"$
\n" ); document.write( "Checking x = 0:
\n" ); document.write( " $\"sqrt%281%2B4%2Asqrt%280%29%29+=+1+%2B+sqrt%280%29\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"sqrt%281%2B4%2A0%29+=+1+%2B+0\"$
\n" ); document.write( " $\"sqrt%281%2B0%29+=+1\"$
\n" ); document.write( " $\"sqrt%281%29+=+1\"$
\n" ); document.write( "1 = 1 Check!

\n" ); document.write( "Checking x = 4:
\n" ); document.write( " $\"sqrt%281%2B4%2Asqrt%284%29%29+=+1+%2B+sqrt%284%29\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"sqrt%281%2B4%2A2%29+=+1+%2B+2\"$
\n" ); document.write( " $\"sqrt%281%2B8%29+=+3\"$
\n" ); document.write( " $\"sqrt%289%29+=+3\"$
\n" ); document.write( "3 = 3 Check!

\n" ); document.write( "So both answers check out. There were no extraneous solutions this time. (But we did not know this until we checked!) \n" ); document.write( "

\n" );