document.write( "Question 263607: Please help me solve this equation: 6x^2-13x-8=0\r
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Algebra.Com's Answer #194258 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
6x^2-13x-8=0
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"6x%5E2-13x-8\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"-8\"$ .

Now multiply the first coefficient $\"6\"$ by the last term $\"-8\"$ to get $\"%286%29%28-8%29=-48\"$ .

Now the question is: what two whole numbers multiply to $\"-48\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?

To find these two numbers, we need to list all of the factors of $\"-48\"$ (the previous product).

Factors of $\"-48\"$ :

1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-48\"$ .

1*(-48) = -48
2*(-24) = -48
3*(-16) = -48
4*(-12) = -48
6*(-8) = -48
(-1)*(48) = -48
(-2)*(24) = -48
(-3)*(16) = -48
(-4)*(12) = -48
(-6)*(8) = -48

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-13\"$ :

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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

From the table, we can see that the two numbers $\"3\"$ and $\"-16\"$ add to $\"-13\"$ (the middle coefficient).

So the two numbers $\"3\"$ and $\"-16\"$ both multiply to $\"-48\"$ and add to $\"-13\"$

Now replace the middle term $\"-13x\"$ with $\"3x-16x\"$ . Remember, $\"3\"$ and $\"-16\"$ add to $\"-13\"$ . So this shows us that $\"3x-16x=-13x\"$ .

$\"6x%5E2%2Bhighlight%283x-16x%29-8\"$ Replace the second term $\"-13x\"$ with $\"3x-16x\"$ .

$\"%286x%5E2%2B3x%29%2B%28-16x-8%29\"$ Group the terms into two pairs.

$\"3x%282x%2B1%29%2B%28-16x-8%29\"$ Factor out the GCF $\"3x\"$ from the first group.

$\"3x%282x%2B1%29-8%282x%2B1%29\"$ Factor out $\"8\"$ 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.

$\"%283x-8%29%282x%2B1%29\"$ Combine like terms. Or factor out the common term $\"2x%2B1\"$

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Answer:

So $\"6%2Ax%5E2-13%2Ax-8\"$ factors to $\"%283x-8%29%282x%2B1%29\"$ .

In other words, $\"6%2Ax%5E2-13%2Ax-8=%283x-8%29%282x%2B1%29\"$ .

Note: you can check the answer by expanding $\"%283x-8%29%282x%2B1%29\"$ to get $\"6%2Ax%5E2-13%2Ax-8\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"6x%5E2%2B-13x%2B-8+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%28-13%29%5E2-4%2A6%2A-8=361\"$ .
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\n" ); document.write( " Discriminant d=361 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28--13%2B-sqrt%28+361+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%28-13%29%2Bsqrt%28+361+%29%29%2F2%5C6+=+2.66666666666667\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%28-13%29-sqrt%28+361+%29%29%2F2%5C6+=+-0.5\"$
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\n" ); document.write( " Quadratic expression $\"6x%5E2%2B-13x%2B-8\"$ can be factored:
\n" ); document.write( " $\"6x%5E2%2B-13x%2B-8+=+6%28x-2.66666666666667%29%2A%28x--0.5%29\"$
\n" ); document.write( " Again, the answer is: 2.66666666666667, -0.5.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-13%2Ax%2B-8+%29\"$

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