document.write( "Question 345301: The instructions on this one say to solve each of the quadratic equations by factoring and applying the property ab=0 if and only if a=0 or b=0 The problem I need help with is:\r
\n" ); document.write( "\n" ); document.write( "4x^2+29x+30=0 \n" ); document.write( "

Algebra.Com's Answer #246924 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll help you factor. I'll let you take over after that.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4x%5E2%2B29x%2B30\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"29\"$ , and the last term is $\"30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"30\"$ to get $\"%284%29%2830%29=120\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"120\"$ (the previous product) and add to the second coefficient $\"29\"$ ?\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 $\"120\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"120\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\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 $\"120\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*120 = 120
\n" ); document.write( "2*60 = 120
\n" ); document.write( "3*40 = 120
\n" ); document.write( "4*30 = 120
\n" ); document.write( "5*24 = 120
\n" ); document.write( "6*20 = 120
\n" ); document.write( "8*15 = 120
\n" ); document.write( "10*12 = 120
\n" ); document.write( "(-1)*(-120) = 120
\n" ); document.write( "(-2)*(-60) = 120
\n" ); document.write( "(-3)*(-40) = 120
\n" ); document.write( "(-4)*(-30) = 120
\n" ); document.write( "(-5)*(-24) = 120
\n" ); document.write( "(-6)*(-20) = 120
\n" ); document.write( "(-8)*(-15) = 120
\n" ); document.write( "(-10)*(-12) = 120\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 $\"29\"$ :\r
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First Number	Second Number	Sum
1	120	1+120=121
2	60	2+60=62
3	40	3+40=43
4	30	4+30=34
5	24	5+24=29
6	20	6+20=26
8	15	8+15=23
10	12	10+12=22
-1	-120	-1+(-120)=-121
-2	-60	-2+(-60)=-62
-3	-40	-3+(-40)=-43
-4	-30	-4+(-30)=-34
-5	-24	-5+(-24)=-29
-6	-20	-6+(-20)=-26
-8	-15	-8+(-15)=-23
-10	-12	-10+(-12)=-22

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"24\"$ add to $\"29\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"24\"$ both multiply to $\"120\"$ and add to $\"29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"29x\"$ with $\"5x%2B24x\"$ . Remember, $\"5\"$ and $\"24\"$ add to $\"29\"$ . So this shows us that $\"5x%2B24x=29x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%285x%2B24x%29%2B30\"$ Replace the second term $\"29x\"$ with $\"5x%2B24x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2%2B5x%29%2B%2824x%2B30%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%284x%2B5%29%2B%2824x%2B30%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%284x%2B5%29%2B6%284x%2B5%29\"$ 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( " $\"%28x%2B6%29%284x%2B5%29\"$ Combine like terms. Or factor out the common term $\"4x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2%2B29x%2B30\"$ factors to $\"%28x%2B6%29%284x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4x%5E2%2B29x%2B30=%28x%2B6%29%284x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B6%29%284x%2B5%29\"$ to get $\"4x%5E2%2B29x%2B30\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider making a donation at my tutoring website. Thank you.\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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