document.write( "Question 757880: 0=4v^2+21v-18\r
\n" ); document.write( "\n" ); document.write( "for this quadratic equation why did they have 0=(4v-3)(v+6) instead of (2v-9)(2v+2)?\r
\n" ); document.write( "\n" ); document.write( "What determines the factors you use if they equal the same thing? 3*6=18 and so does 2*9. I don't get why it was wrong. \n" ); document.write( "

Algebra.Com's Answer #461332 by KMST(5328) $\"\"$ $\"About$

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Factoring the quadratic polynomial $\"4v%5E2%2B21v-18\"$ is one way of solving the equation $\"4v%5E2%2B21v-18=0\"$ . Factoring works well for this equation, if not easily.
\n" ); document.write( " $\"%282v-9%29%282v%2B2%29=4v%5E2%2B4v-18v-18=4v%5E2-14v-18\"$ is not a factorization of $\"4v%5E2%2B21v-18\"$
\n" ); document.write( " $\"%284v-3%29%28v%2B6%29=4v%5E2%2B24v-3v-18=4v%5E2%2B21v-18\"$ is the factorization of $\"4v%5E2%2B21v-18\"$
\n" ); document.write( "Not all polynomials can be factorized.
\n" ); document.write( "If there is a simple factorization, it is easier if the polynomial starts with $\"x%5E2\"$ .
\n" ); document.write( "If there is a leading coefficient in front of $\"x%5E2\"$ , factoring gets complicated, but there is a method to find the four coefficients of the factorization.
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\n" ); document.write( "THE IDEA:
\n" ); document.write( "When you multiply $\"%282v-9%29%282v%2B2%29\"$ or $\"%284v-3%29%28v%2B6%29\"$ you get a sum of all possible products (4 products). Teachers tell students to use the acronym \"FOIL\" to make sure they include the right four products, and do not get mixed up.
\n" ); document.write( "They tell you to \"FOIL\", meaning to write the sum of the four terms. Then you have to add together the two terms in x. They call that \"collect like terms\".
\n" ); document.write( "To factor you have to \"un-FOIL\", reversing the procedure to get the four \"FOIL\" terms.
\n" ); document.write( "You know that of the four products that make $\"4v%5E2%2B21v-18\"$ , the first is $\"4v%5E2\"$ and the last is $\"-18\"$ .
\n" ); document.write( "The trick is to find the other two terms that get collected together to form $\"21v\"$ .
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\n" ); document.write( "THE PROCEDURE:
\n" ); document.write( "Multiply the $\"4\"$ from $\"4v%5E2\"$ and the $\"-18\"$ .
\n" ); document.write( "You get a product, $\"4%2A%28-18%29=-72\"$ , that is also the product of the missing terms' coefficients.
\n" ); document.write( "You want to write $\"-72\"$ as the product of two factors thaT ADD UP TO THE $\"21\"$ in $\"21v\"$ .
\n" ); document.write( "Not worrying (for now) about the minus sign, the ways to write $\"72\"$ as a product are:
\n" ); document.write( " $\"1%2A72=72\"$
\n" ); document.write( " $\"2%2A36=72\"$
\n" ); document.write( " $\"3%2A24=72\"$
\n" ); document.write( " $\"4%2A18=72\"$
\n" ); document.write( " $\"6%2A12=72\"$ and
\n" ); document.write( " $\"8%2A9=72\"$
\n" ); document.write( "To multiply to $\"-72\"$ , we need to give one factor a minus sign.
\n" ); document.write( "To add to $\"21\"$ we need to give the minus sign to the smaller factor, and we need oine factor much larger than the other.
\n" ); document.write( " $\"18-4=14\"$ does not work. $\"18\"$ and $\"4\"$ are not far apart enough.
\n" ); document.write( " $\"24-3=21\"$ is the solution.
\n" ); document.write( "The FOIL would be $\"4v%5E2%2B24v-3v-18\"$ .
\n" ); document.write( "We write those terms in a tic-tac-toe table, like this:
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\n" ); document.write( "Then we write the common factors for each row and column:
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\n" ); document.write( "The first row and the first column are the factors $\"%28v%2B6%29\"$ , and $\"%284v-3%29\"$ \n" ); document.write( "

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