document.write( "Question 151702: Solve 45x^2-60x+20=0 \n" ); document.write( "

Algebra.Com's Answer #111518 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"45x%5E2-60x%2B20\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"5%289x%5E2-12x%2B4%29\"$ Factor out the GCF $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"9x%5E2-12x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2-12x%2B4\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"4\"$ to get $\"%289%29%284%29=36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?\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 $\"36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"36\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\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 $\"36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*36
\n" ); document.write( "2*18
\n" ); document.write( "3*12
\n" ); document.write( "4*9
\n" ); document.write( "6*6
\n" ); document.write( "(-1)*(-36)
\n" ); document.write( "(-2)*(-18)
\n" ); document.write( "(-3)*(-12)
\n" ); document.write( "(-4)*(-9)
\n" ); document.write( "(-6)*(-6)\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 $\"-12\"$ :\r
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First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-6\"$ both multiply to $\"36\"$ and add to $\"-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-12x\"$ with $\"-6x-6x\"$ . Remember, $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ . So this shows us that $\"-6x-6x=-12x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%28-6x-6x%29%2B4\"$ Replace the second term $\"-12x\"$ with $\"-6x-6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2-6x%29%2B%28-6x%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-2%29%2B%28-6x%2B4%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-2%29-2%283x-2%29\"$ Factor out $\"2\"$ 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( " $\"%283x-2%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-2%29%5E2\"$ Condense the common factors.\r
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\n" ); document.write( "\n" ); document.write( "So the expression goes from $\"5%289x%5E2-12x%2B4%29\"$ and factors further to $\"5%283x-2%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"45x%5E2-60x%2B20\"$ factors to $\"5%283x-2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5%283x-2%29%5E2=0\"$ Set the factored expression equal to zero\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-2%29%5E2=0\"$ Divide both sides by 5.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x-2=0\"$ Take the square root of both sides\r
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\n" ); document.write( "\n" ); document.write( " $\"3x=0%2B2\"$ Add $\"2\"$ to both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x=2\"$ Combine like terms on the right side.\r
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\n" ); document.write( "\n" ); document.write( " $\"x=%282%29%2F%283%29\"$ Divide both sides by $\"3\"$ to isolate $\"x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So the answer is $\"x=2%2F3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Which approximates to $\"x=0.667\"$ \r
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