document.write( "Question 263255: solve: radical 6x-2=radical 2x+3 - radical 4x-1\r
\n" ); document.write( "\n" ); document.write( "i did not know what to use for the radical sign? is it ~ ? thank you. \n" ); document.write( "

Algebra.Com's Answer #194205 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%286x-2%29+=+sqrt%282x%2B3%29+-+sqrt%284x-1%29\"$
\n" ); document.write( "The basic process for solving equations with variables inside square roots is:

Solve the equation for one of the square roots (i.e. get one square root by itself on one side of the equation).
Square both sides of the equation. This eliminates the \"solved for\" square root.
If there are still square roots remaining, repeat steps #1 and #2.
At this point there should no longer be any square roots. Solve this equation using the appropriate steps for the kind of equation you now have.
Check your answer(s). This is not just a good idea. We squared both sides of the equation at least once (step #2). And squaring both sides of an equation may introduce what are called extraneous solutions. Extraneous solutions are solutions which work in the squared equation but do not work in the original equation. Extraneous solutions must be identified and rejected in the checking process. (Do not use an equation you have squared to check your answer(s)!)

\n" ); document.write( "Let's see how this works on your equation:
\n" ); document.write( "1) Solve for one of the square roots. Your equation already has a square root by itself on the left side of the equation.

\n" ); document.write( "2) Square both sides:
\n" ); document.write( " $\"%28sqrt%286x-2%29%29%5E2+=+%28sqrt%282x%2B3%29+-+sqrt%284x-1%29%29%5E2\"$
\n" ); document.write( "Squaring the left side is simple. Squaring the right side requires care. There are two terms on the right side. To square the right side we need to use FOIL or one of the patterns:

$\"%28a%2Bb%29%5E2+=+a%5E2+%2B+2ab+%2B+b%5E2\"$
$\"%28a-b%29%5E2+=+a%5E2+-+2ab+%2B+b%5E2\"$

\n" ); document.write( "Squaring we get:
\n" ); document.write( "

\n" ); document.write( "Simplifying:
\n" ); document.write( " $\"6x-2+=+2x%2B3+-2sqrt%282x%2B3%29%2A%284x-1%29%29+%2B+4x-1\"$
\n" ); document.write( " $\"6x-2+=+6x%2B2+-2sqrt%288x%5E2+%2B+10x+-+3%29\"$

\n" ); document.write( "3) We still have a square root. So we have to repeat steps #1 and #2:
\n" ); document.write( "Solve for a square root. Subtract 6x from each side:
\n" ); document.write( " $\"-2+=+2+-2sqrt%288x%5E2+%2B+10x+-+3%29\"$
\n" ); document.write( "Subtract 2 from each side:
\n" ); document.write( " $\"-4+=+-2sqrt%288x%5E2+%2B+10x+-+3%29\"$
\n" ); document.write( "Divide both sides by -2:
\n" ); document.write( " $\"2+=+sqrt%288x%5E2+%2B+10x+-+3%29\"$
\n" ); document.write( "The square root on the right is \"solved for\". Square both sides:
\n" ); document.write( " $\"%282%29%5E2+=+%28sqrt%288x%5E2+%2B+10x+-+3%29%29%5E2\"$
\n" ); document.write( " $\"4+=+8x%5E2+%2B+10x+-+3\"$

\n" ); document.write( "4) We have finally eliminated all the square roots and we can proceed to solving the equation we have. This is a quadratic equation (because of the $\"x%5E2\"$ ) so we will get one side equal to zero (by subtracting 4):
\n" ); document.write( " $\"0+=+8x%5E2+%2B+10x+-+7\"$
\n" ); document.write( "Next we factor the non-zero side or use the Quadratic Formula. This factors fairly easily:
\n" ); document.write( " $\"0+=+%284x+%2B+7%29%282x+-+1%29\"$
\n" ); document.write( "Now we can use the Zero Product Property which tells us that this product can be zero only if one of the factors is zero. So:
\n" ); document.write( " $\"4x+%2B+7+=+0\"$ or $\"2x+-+1+=+0\"$
\n" ); document.write( "Solving these we get:
\n" ); document.write( " $\"x+=+-7%2F4\"$ or $\"x+=+1%2F2\"$

\n" ); document.write( "5) Check the solution(s). (Remember, this is not optional because we squared both sides of the equation.) Using the original equation:
\n" ); document.write( " $\"sqrt%286x-2%29+=+sqrt%282x%2B3%29+-+sqrt%284x-1%29\"$
\n" ); document.write( "Checking -7/4:
\n" ); document.write( " $\"sqrt%286%28-7%2F4%29-2%29+=+sqrt%282%28-7%2F4%29%2B3%29+-+sqrt%284%28-7%2F4%29-1%29\"$
\n" ); document.write( " $\"sqrt%28-42%2F4-2%29+=+sqrt%28-14%2F4%2B3%29+-+sqrt%28-7-1%29\"$
\n" ); document.write( " $\"sqrt%28-42%2F4-8%2F4%29+=+sqrt%28-14%2F4%2B12%2F4%29+-+sqrt%28-7-1%29\"$
\n" ); document.write( " $\"sqrt%28-50%2F4%29+=+sqrt%28-2%2F4%29+-+sqrt%28-8%29\"$
\n" ); document.write( "As we can see, all three radicands (\"radicand\": the expression within a radical) are negative. We cannot allow negatives in square roots. So we must reject -7/4 as a solution. (Note: If even only one radicand was negative, we would still have to reject this solution.)

\n" ); document.write( "Checking 1/2:
\n" ); document.write( " $\"sqrt%286%281%2F2%29-2%29+=+sqrt%282%281%2F2%29%2B3%29+-+sqrt%284%281%2F2%29-1%29\"$
\n" ); document.write( " $\"sqrt%283-2%29+=+sqrt%281%2B3%29+-+sqrt%282-1%29\"$
\n" ); document.write( " $\"sqrt%281%29+=+sqrt%284%29+-+sqrt%281%29\"$
\n" ); document.write( " $\"1+=+2+-+1%29\"$
\n" ); document.write( " $\"1+=+1\"$ Check!!

\n" ); document.write( "So the only solution to
\n" ); document.write( " $\"sqrt%286x-2%29+=+sqrt%282x%2B3%29+-+sqrt%284x-1%29\"$
\n" ); document.write( "is x = 1/2. \n" ); document.write( "

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