document.write( "Question 384708: \n" ); document.write( "

Algebra.Com's Answer #272372 by jsmallt9(3758) $\"\"$ $\"About$

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$\"sqrt%282x%29-sqrt%28x-1%29-1+=+0\"$
\n" ); document.write( "Solving equations where the variable is inside one or more square roots can be done by:

Isolate a square root.
Square both sides of the equation.
If there are any remaining square roots, repeat steps 1 and 2.
At this point there should be no square roots remaining. Solve the equation using appropriate techniques for the type of equation.
Check your answer(s)! This is not optional. Squaring both sides of an equation (which has been done at least once) can introduce what are called extraneous solutions. Extraneous solutions are solutions that fit the squared equation but do not fit the original equation! Extraneous solutions can occur even if no mistakes have been made. So you must check your answers and extraneous solutions, if any, must be rejected. (It is even possible that all \"solutions\" are extraneous meaning that all \"solutions\" are rejected leaving no solution to your equation!)

\n" ); document.write( "Let's see this in action:
\n" ); document.write( "1) Isolate a square root.
\n" ); document.write( "It doesn't matter which square root we isolate. We can isolate the second square root by adding it to both sides:
\n" ); document.write( " $\"sqrt%282x%29-1+=+sqrt%28x-1%29\"$
\n" ); document.write( "2) Square both sides:
\n" ); document.write( " $\"%28sqrt%282x%29-1%29%5E2+=+%28sqrt%28x-1%29%29%5E2\"$
\n" ); document.write( "The right side simplifies easily. But we have to be careful squareing the left side. Exponents do not distribute! We must use FOIL or the patter $\"%28a-b%29%5E2+=+a%5E2+-2ab+%2B+b%5E2\"$ . I like using the pattern:
\n" ); document.write( " $\"%28sqrt%282x%29%29%5E2+-+2%28sqrt%282x%29%29%281%29+%2B+1%5E2+=+x-1\"$
\n" ); document.write( " $\"2x+-+2sqrt%282x%29+%2B+1+=+x-1\"$
\n" ); document.write( "3) There is still a square root so we must repeat steps 1 and 2.
\n" ); document.write( "Isolate a square root.
\n" ); document.write( "Subtracting 2x and 1 from each side we get:
\n" ); document.write( " $\"-2sqrt%282x%29+=+-x-2\"$
\n" ); document.write( "Square both sides:
\n" ); document.write( " $\"%28-2sqrt%282x%29%29%5E2+=+%28-x-2%29%5E2\"$
\n" ); document.write( " $\"4%2A2x+=+%28-x%29%5E2+-2%28-x%29%282%29+%2B+%28-2%29%5E2\"$
\n" ); document.write( " $\"8x+=+x%5E2+%2B4x+%2B+4\"$
\n" ); document.write( "4) Solve the equation.
\n" ); document.write( "This appears to be a quadratic equation. So we want one side to be zero. Subtracting 8x from each side we get:
\n" ); document.write( " $\"0+=+x%5E2+-4x+%2B+4\"$
\n" ); document.write( "Factoring we get:
\n" ); document.write( "0 = (x-2)(x-2)
\n" ); document.write( "Using the Zero Product Property which tells us that this (or any) product can be zero only if one (or more) of the factors is zero. So:
\n" ); document.write( "x-2 = 0
\n" ); document.write( "Solving this we get
\n" ); document.write( "x = 2
\n" ); document.write( "5) Check you answer(s).
\n" ); document.write( "Always us the original equation to check:
\n" ); document.write( " $\"sqrt%282x%29-sqrt%28x-1%29-1+=+0\"$
\n" ); document.write( "Checking x = 2:
\n" ); document.write( " $\"sqrt%282%282%29%29-sqrt%28%282%29-1%29-1+=+0\"$
\n" ); document.write( " $\"sqrt%284%29-sqrt%281%29-1+=+0\"$
\n" ); document.write( "2 - 1 - 1 = 0
\n" ); document.write( "0 = 0 Check! \n" ); document.write( "

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