document.write( "Question 232351: solve x using quadratic formula:
\n" ); document.write( "p2x2+(p2-q2)x-q2=0 \n" ); document.write( "

Algebra.Com's Answer #171889 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The general form of the quadratic equation is:
\n" ); document.write( " $\"ax%5E2%2Bbx%2Bc=0\"$
\n" ); document.write( "Your equation is
\n" ); document.write( " $\"p%5E2x%5E2+%2B+%28p%5E2-q%5E2%29x-q%5E2=0\"$
\n" ); document.write( "or
\n" ); document.write( " $\"%28p%5E2%29x%5E2+%2B+%28p%5E2-q%5E2%29x+%2B+%28-q%5E2%29=0\"$

\n" ); document.write( "If we think of the \"a\", \"b\" and \"c\" as place-holders, we should be able to see your equation as a quadratic with $\"p%5E2\"$ for \"a\", $\"%28p%5E2-q%5E2%29\"$ for \"b\" and $\"-q%5E2\"$ for \"c\". And we can substitute these into the quadratic formula,
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28b%5E2+-4ac%29%29%2F%282a%29\"$ giving us:
\n" ); document.write( "

\n" ); document.write( "Now we just simplify as much as possible:
\n" ); document.write( "

\n" ); document.write( " $\"x+=+%28-p%5E2%2Bq%5E2+%2B-+sqrt%28p%5E4%2B+2p%5E2q%5E2+%2B+q%5E4%29%29%2F%282p%5E2%29\"$
\n" ); document.write( " $\"x+=+%28-p%5E2%2Bq%5E2+%2B-+sqrt%28%28p%5E2%2B+q%5E2%29%5E2%29%29%2F%282p%5E2%29\"$
\n" ); document.write( " $\"x+=+%28-p%5E2%2Bq%5E2+%2B-+%28p%5E2%2B+q%5E2%29%29%2F%282p%5E2%29\"$
\n" ); document.write( "(Note: Normally $\"sqrt%28x%5E2%29+=+abs%28x%29\"$ . But since $\"p%5E2+%2B+q%5E2\"$ must be positive, no matter what p and q are, we do not need to be concerned about absolute value.)
\n" ); document.write( "Now that we have reached the \"+-\", we rewrite this as two equations:
\n" ); document.write( " $\"x+=+%28-p%5E2%2Bq%5E2+%2B+%28p%5E2%2B+q%5E2%29%29%2F%282p%5E2%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"x+=+%28-p%5E2%2Bq%5E2+-+%28p%5E2%2B+q%5E2%29%29%2F%282p%5E2%29\"$
\n" ); document.write( "And we will finish simplifying them separately.
\n" ); document.write( " $\"x+=+%282q%5E2%29%2F%282p%5E2%29+=+q%5E2%2Fp%5E2\"$
\n" ); document.write( "or
\n" ); document.write( " $\"x+=+%28-2p%5E2%29%2F%282p%5E2%29+=+-1\"$ \n" ); document.write( "

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