document.write( "Question 834167: Solve. try facoting first, it factoring is not possible or is difficult, use the quadratic formula.
\n" ); document.write( "2t^2-3t=1 \n" ); document.write( "

Algebra.Com's Answer #502845 by KMST(5328) $\"\"$ $\"About$

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$\"2t%5E2-3t=1\"$
\n" ); document.write( " $\"2t%5E2-3t-1=0\"$
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\n" ); document.write( "FACTORING:
\n" ); document.write( "We multiply the $\"2\"$ that is in front of $\"t%5E2\"$ times the $\"-1\"$ at the end and get
\n" ); document.write( " $\"2%2A%28-1%29=red%28-2%29\"$ .
\n" ); document.write( "Now, we look for pairs of factors of $\"red%28-2%29\"$ that will ad up to the $\"-3\"$ in front of $\"t\"$ .
\n" ); document.write( "We cannot find any pair that works.
\n" ); document.write( "We find $\"2%2A%28-1%29=red%28-2%29\"$ and $\"1%2A%28-2%29=red%28-2%29\"$ but the sums are wrong:
\n" ); document.write( " $\"2%2B%28-1%29=1\"$ and $\"1%2B%28-2%29=-1\"$ .
\n" ); document.write( "If the coefficient in front of $\"t\"$ was $\"1\"$ or $\"-1\"$ ,
\n" ); document.write( "we could solve $\"2t%5E2%2Bt-1=0\"$ or $\"2t%5E2-t-1=0\"$ by factoring, but
\n" ); document.write( "we cannot solve $\"2t%5E2-3t-1=0\"$ <--> $\"2t%5E2-3t=1\"$ by factoring.
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\n" ); document.write( "APPLYING THE QUADRATIC FORMULA:
\n" ); document.write( "The quadratic formula says that the solutions to $\"ax%5E2%2Bbx%2Bc=0\"$ are given by
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ .
\n" ); document.write( "In $\"2t%5E2-3t-1=0\"$ we have $\"t\"$ instead of $\"x\"$ ,
\n" ); document.write( "but the name of the variable does not matter.
\n" ); document.write( " $\"a=2\"$ is the coefficient of the term with the variable squared.
\n" ); document.write( " $\"b=-3\"$ is the coefficient of the term with the variable not squared.
\n" ); document.write( " $\"c=-1\"$ is the independent term.
\n" ); document.write( "Applying the formula with those values substituted (and $\"t\"$ instead of $\"x\"$ , of course) we get
\n" ); document.write( " $\"t+=+%28-%28-3%29+%2B-+sqrt%28%28-3%29%5E2-4%2A2%2A%28-1%29%29%29%2F%282%2A2%29+\"$
\n" ); document.write( " $\"t+=+%283+%2B-+sqrt%289%2B8%29%29%2F4+\"$
\n" ); document.write( " $\"highlight%28t+=+%283+%2B-+sqrt%2817%29%29%2F4%29\"$
\n" ); document.write( "That is an ugly expression that says that the solutions are the irrational numbers
\n" ); document.write( " $\"%283+%2B+sqrt%2817%29%29%2F4=approximately\"$ $\"1.78\"$
\n" ); document.write( "and
\n" ); document.write( " $\"%283+-+sqrt%2817%29%29%2F4=approximately\"$ $\"-0.28\"$ .
\n" ); document.write( "No wonder factoring did not work.
\n" ); document.write( "Factoring can only work when the solutions are rational numbers. \n" ); document.write( "

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