document.write( "Question 873910: Factor the following quadric equation: x^2+x-30 \n" ); document.write( "

Algebra.Com's Answer #527182 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2Bx-30\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-30\"$ to get $\"%281%29%28-30%29=-30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"1\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-30\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-30) = -30
\n" ); document.write( "2*(-15) = -30
\n" ); document.write( "3*(-10) = -30
\n" ); document.write( "5*(-6) = -30
\n" ); document.write( "(-1)*(30) = -30
\n" ); document.write( "(-2)*(15) = -30
\n" ); document.write( "(-3)*(10) = -30
\n" ); document.write( "(-5)*(6) = -30\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"1\"$ :\r
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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"6\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"6\"$ both multiply to $\"-30\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1x\"$ with $\"-5x%2B6x\"$ . Remember, $\"-5\"$ and $\"6\"$ add to $\"1\"$ . So this shows us that $\"-5x%2B6x=1x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-5x%2B6x%29-30\"$ Replace the second term $\"1x\"$ with $\"-5x%2B6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-5x%29%2B%286x-30%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-5%29%2B%286x-30%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-5%29%2B6%28x-5%29\"$ Factor out $\"6\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B6%29%28x-5%29\"$ Combine like terms. Or factor out the common term $\"x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2Bx-30\"$ factors to $\"%28x%2B6%29%28x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2Bx-30=%28x%2B6%29%28x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B6%29%28x-5%29\"$ to get $\"x%5E2%2Bx-30\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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