document.write( "Question 912772: factor complete : 7x^2 + 33x - 10 = 0 \n" ); document.write( "

Algebra.Com's Answer #554030 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at the expression $\"7x%5E2%2B33x-10\"$ , we can see that the first coefficient is $\"7\"$ , the second coefficient is $\"33\"$ , and the last term is $\"-10\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"7\"$ by the last term $\"-10\"$ to get $\"%287%29%28-10%29=-70\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-70\"$ (the previous product) and add to the second coefficient $\"33\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-70\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-70\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,7,10,14,35,70\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-7,-10,-14,-35,-70\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-70\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-70) = -70
\n" ); document.write( "2*(-35) = -70
\n" ); document.write( "5*(-14) = -70
\n" ); document.write( "7*(-10) = -70
\n" ); document.write( "(-1)*(70) = -70
\n" ); document.write( "(-2)*(35) = -70
\n" ); document.write( "(-5)*(14) = -70
\n" ); document.write( "(-7)*(10) = -70\r
\n" ); document.write( "
\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 $\"33\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-70	1+(-70)=-69
2	-35	2+(-35)=-33
5	-14	5+(-14)=-9
7	-10	7+(-10)=-3
-1	70	-1+70=69
-2	35	-2+35=33
-5	14	-5+14=9
-7	10	-7+10=3

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"35\"$ add to $\"33\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"35\"$ both multiply to $\"-70\"$ and add to $\"33\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"33x\"$ with $\"-2x%2B35x\"$ . Remember, $\"-2\"$ and $\"35\"$ add to $\"33\"$ . So this shows us that $\"-2x%2B35x=33x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"7x%5E2%2Bhighlight%28-2x%2B35x%29-10\"$ Replace the second term $\"33x\"$ with $\"-2x%2B35x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%287x%5E2-2x%29%2B%2835x-10%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%287x-2%29%2B%2835x-10%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%287x-2%29%2B5%287x-2%29\"$ Factor out $\"5\"$ 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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%29%287x-2%29\"$ Combine like terms. Or factor out the common term $\"7x-2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"7x%5E2%2B33x-10\"$ factors to $\"%28x%2B5%29%287x-2%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"7x%5E2%2B33x-10=%28x%2B5%29%287x-2%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B5%29%287x-2%29\"$ to get $\"7x%5E2%2B33x-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Let me know if you need more help or if you need me to explain a step in more detail.
\n" ); document.write( "Feel free to email me at jim_thompson5910@hotmail.com
\n" ); document.write( "or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thanks,\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim
\n" ); document.write( " \n" ); document.write( "

\n" );