document.write( "Question 261529: how is the following equation solved\r
\n" ); document.write( "\n" ); document.write( "z - 7z1/2 + 12 = 0 \n" ); document.write( "

Algebra.Com's Answer #192692 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"z+-+7z%5E%281%2F2%29+%2B+12+=+0\"$
\n" ); document.write( "The key to solving this equation is to recognize that $\"z+=+%28z%5E%281%2F2%29%29%5E2\"$ . This is what makes this a quadratic equation. If you have trouble seeing this, then we can use a temporary variable:
\n" ); document.write( "Let $\"q+=+z%5E%281%2F2%29\"$
\n" ); document.write( "then $\"q%5E2+=+z\"$
\n" ); document.write( "Replacing z and $\"sqrt%28z%29\"$ with $\"q%5E2\"$ and q respectively your equation becomes:
\n" ); document.write( " $\"q%5E2+-7q+%2B+12+=+0\"$
\n" ); document.write( "This is clearly a quadratic equation. We can solve this by factoring:
\n" ); document.write( " $\"%28q-3%29%28q-4%29+=+0\"$
\n" ); document.write( "From the Zero Product Property we know that this product can be zero only if one of the factors is zero. So:
\n" ); document.write( " $\"q-3+=+0\"$ or $\"q-4+=+0\"$
\n" ); document.write( "Solving these we get:
\n" ); document.write( " $\"q+=+3\"$ or $\"q+=+4\"$

\n" ); document.write( "Of course we are not interested in what q is. We want a solution for z. So at this point we replace q with $\"sqrt%28z%29\"$ :
\n" ); document.write( " $\"z%5E%281%2F2%29+=+3\"$ or $\"z%5E%281%2F2%29+=+4\"$
\n" ); document.write( "We now have one more step. We need to square both sides of these equations giving us:
\n" ); document.write( " $\"z+=+9\"$ or $\"z+=+16\"$

\n" ); document.write( "Using a temporary variable is not required. But they can be helpful until you get used to working without them. Here is a solution which does not use a temporary variable:
\n" ); document.write( " $\"z+-+7z%5E%281%2F2%29+%2B+12+=+0\"$
\n" ); document.write( "Factor:
\n" ); document.write( " $\"%28z%5E%281%2F2%29+-+3%29%28z%5E%281%2F2%29+-+4%29+=+0\"$
\n" ); document.write( "Zero Product Property:
\n" ); document.write( " $\"z%5E%281%2F2%29+-+3+=+0\"$ or $\"z%5E%281%2F2%29+-+4+=+0\"$
\n" ); document.write( "Solve:
\n" ); document.write( " $\"z%5E%281%2F2%29+=+3\"$ or $\"z%5E%281%2F2%29+=+4\"$
\n" ); document.write( " $\"z+=+9\"$ or $\"z+=+16\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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