document.write( "Question 598048: please help me solve this equation using the means and extremes. factor this. $\"+8x%5E2%2B13x-6+\"$ \n" ); document.write( "

Algebra.Com's Answer #378598 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The expression \"means and extremes\" is new to me, but it may be a name for a factoring trick that high school tried to teach my youngest son, so I will show you that.
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\n" ); document.write( "THE TRICK:
\n" ); document.write( "You take the coefficients at both ends and multiply them, as in
\n" ); document.write( " $\"8%2A6=48\"$
\n" ); document.write( "(you do not need to worry about negative signs).
\n" ); document.write( "Then, you look for pairs factors of that product number. For 48, you find:
\n" ); document.write( " $\"48=1%2A48\"$
\n" ); document.write( " $\"48=2%2A24\"$
\n" ); document.write( " $\"48=3%2A16\"$
\n" ); document.write( " $\"48=4%2A12\"$
\n" ); document.write( " $\"48=6%2A8\"$
\n" ); document.write( "Next, you find the pair of factors whose sum or difference is equal to the coefficient of x (13 in your case). You use difference if the independent term is negative, as the -6 in your polynomial.
\n" ); document.write( "In this case, I try to get 13 by doing
\n" ); document.write( " $\"48-1=47\"$
\n" ); document.write( " $\"24-2=22\"$
\n" ); document.write( " $\"16-3=13\"$
\n" ); document.write( "I wrote them all out, but you would do these calculations in your head (or on scrap paper), and do as few as possible.
\n" ); document.write( "For the next step, you draw a large tic-tac-toe grid and enter $\"8x%5E2\"$ in the center and $\"-6\"$ in the bottom right space. Enter $\"16x\"$ and $\"-3x\"$ in the spaces in between the $\"8x%5E2\"$ and $\"-6\"$ filling a 2X2 square.
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(It really does not matter in what corner you place the $\"16x\"$ and the $\"-3x\"$ . I could swap their places, and it would work just as well).
\n" ); document.write( "In the spaces above and to the left of those you filled, write the greatest common factor for the row or column.
\n" ); document.write( "You are making a multiplication table, where the expressions you already entered are the product of the expressions at the head of the corresponding row and column.
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\n" ); document.write( "You would get $\"8x\"$ as the greatest common factor for $\"8x%5E2\"$ and $\"16x\"$ .
\n" ); document.write( "You would get $\"x\"$ as the greatest common factor for $\"8x%5E2\"$ and $\"-3x\"$ .
\n" ); document.write( "You would get $\"2\"$ as the greatest common factor for $\"-6\"$ and $\"16x\"$ ,
\n" ); document.write( "and that would leave $\"-3\"$ as the greatest common factor for $\"-6\"$ and $\"-3x\"$ .
\n" ); document.write( "The expressions in the first row are one of your factors.
\n" ); document.write( "The expressions in the first column are the other factor.
\n" ); document.write( "Your factors are
\n" ); document.write( " $\"%28x%2B2%29%288x-3%29=8x%5E2%2B13x-6\"$
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\n" ); document.write( "THE RATIONALE:
\n" ); document.write( "If a quadratic trinomial in x (such as the $\"8x%5E2%2B13x-6\"$ polynomial in your problem) can be factored into something of the form
\n" ); document.write( " $\"%28ax%2Bb%29%28cx%2Bd%29\"$ with integer coefficients a, b, c, and d,
\n" ); document.write( " $\"%28ax%2Bb%29%28cx%2Bd%29=acx%5E2%2B%28ad%2Bbc%29x%2Bbd\"$ so the coefficient of the term in x, in the middle of the polynomial,
\n" ); document.write( " $\"ad%2Bbc\"$ , can be written as the sum of two numbers whose product is
\n" ); document.write( " $\"%28ac%29%2A%28bd%29\"$ , which is the product of the other two coefficients, the coefficient of $\"x%5E2\"$ , $\"ac\"$ , and the independent term or constant, $\"bd\"$ . \n" ); document.write( "

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