document.write( "Question 613983: x+√x+5=7 \n" ); document.write( "

Algebra.Com's Answer #386343 by jsmallt9(3758) $\"\"$ $\"About$

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$\"x+%2B+sqrt%28x%29+%2B+5+=+7\"$
\n" ); document.write( "The easy way, IMHO, to solve this is to recognize that since the exponent of x, 1, is twice the exponent of $\"sqrt%28x%29\"$ , 1/2, this equation is in what is called \"quadratic form\". Quadratic form equations can be solved like quadratic equations.

\n" ); document.write( "If you are new to equations of quadratic form equations, then it can be very helpful to use a temporary variable:
\n" ); document.write( "Let q = $\"sqrt%28x%29\"$
\n" ); document.write( "Then $\"q%5E2+=+%28sqrt%28x%29%29%5E2+=+x\"$
\n" ); document.write( "Substituting these into our equation we get:
\n" ); document.write( " $\"q%5E2+%2B+q+%2B+5+=+7\"$
\n" ); document.write( "This is obviously a quadratic equation. To solve, we first make one side equal to zero. Subtracting 7 from each side we get:
\n" ); document.write( " $\"q%5E2+%2B+q+-+2+=+0\"$
\n" ); document.write( "Then we factor:
\n" ); document.write( "(q + 2)(q - 1) = 0
\n" ); document.write( "One of these factors must be zero:
\n" ); document.write( "q + 2 = 0 or q - 1 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "q = -2 or q = 1

\n" ); document.write( "But we are not interested in what values \"q\" might be. We want to know values for \"x\". So we substitute back in for the q's:
\n" ); document.write( " $\"sqrt%28x%29+=+-2\"$ or $\"sqrt%28x%29+=+1\"$
\n" ); document.write( "Since square roots are never negative there is no solution to the first equation. But we will get solution from the second equation. Squaring both sides we get:
\n" ); document.write( "x = 1

\n" ); document.write( "Since we squared both sides, we must check our solution. Use the original equation to check:
\n" ); document.write( " $\"x+%2B+sqrt%28x%29+%2B+5+=+7\"$
\n" ); document.write( "Checking x = 1:
\n" ); document.write( " $\"%281%29+%2B+sqrt%28%281%29%29+%2B+5+=+7\"$
\n" ); document.write( " $\"1+%2B+1+%2B+5+=+7\"$
\n" ); document.write( "7 = 7 Check!

\n" ); document.write( "After a few of these quadratic form equations, you will no longer need a temporary variable. You will see that
\n" ); document.write( " $\"x+%2B+sqrt%28x%29+-+2+=+0\"$
\n" ); document.write( "will factor into:
\n" ); document.write( " $\"sqrt%28x%29+%2B+2%29%28sqrt%28x%29-1%29+=+0\"$
\n" ); document.write( "etc.

\n" ); document.write( "An alternate way to solve this is as a square root equation:
\n" ); document.write( " $\"x+%2B+sqrt%28x%29+%2B+5+=+7\"$
\n" ); document.write( "1. Isolate a the square root
\n" ); document.write( "Subtracting x and 5 from each side:
\n" ); document.write( " $\"sqrt%28x%29+=+2-x+\"$
\n" ); document.write( "2. Square both sides:
\n" ); document.write( " $\"%28sqrt%28x%29%29%5E2+=+%282-x%29%5E2+\"$
\n" ); document.write( " $\"x+=+4+-+4x+%2B+x%5E2\"$
\n" ); document.write( "3. Solve the resulting equation.
\n" ); document.write( "This is quadratic so we want one side to be zero, Subtracting x from each side:
\n" ); document.write( " $\"0+=+4+-+5x+%2B+x%5E2\"$
\n" ); document.write( "4. Factor (or use the Quadratic Formula):
\n" ); document.write( "(x - 4)(x - 1) = 0
\n" ); document.write( "5. One factor must be zero:
\n" ); document.write( "x - 4 = 0 or x - 1 = 0
\n" ); document.write( "6. Solve
\n" ); document.write( "x = 4 or x = 1
\n" ); document.write( "7. Check. (Again, we squared both sides earlier so the check is not optional!)
\n" ); document.write( " $\"x+%2B+sqrt%28x%29+%2B+5+=+7\"$
\n" ); document.write( "Checking x = 4:
\n" ); document.write( " $\"%284%29+%2B+sqrt%28%284%29%29+%2B+5+=+7\"$
\n" ); document.write( "4 + 2 + 5 = 7
\n" ); document.write( "11 = 7 Check failed! Reject x = 4
\n" ); document.write( "Checking x = 1:
\n" ); document.write( " $\"%281%29+%2B+sqrt%28%281%29%29+%2B+5+=+7\"$
\n" ); document.write( " $\"1+%2B+1+%2B+5+=+7\"$
\n" ); document.write( "7 = 7 Check! \n" ); document.write( "

\n" );