document.write( "Question 908439: Please, help me to solve the quadratic equation: 5x²-8=18x \n" ); document.write( "

Algebra.Com's Answer #551028 by richwmiller(17219) $\"\"$ $\"About$

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5x²+18x-8
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"5x%5E2%2B18x-8\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"18\"$ , and the last term is $\"-8\"$ .

Now multiply the first coefficient $\"5\"$ by the last term $\"-8\"$ to get $\"%285%29%28-8%29=-40\"$ .

Now the question is: what two whole numbers multiply to $\"-40\"$ (the previous product) and add to the second coefficient $\"18\"$ ?

To find these two numbers, we need to list all of the factors of $\"-40\"$ (the previous product).

Factors of $\"-40\"$ :

1,2,4,5,8,10,20,40

-1,-2,-4,-5,-8,-10,-20,-40

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

These factors pair up and multiply to $\"-40\"$ .

1*(-40) = -40
2*(-20) = -40
4*(-10) = -40
5*(-8) = -40
(-1)*(40) = -40
(-2)*(20) = -40
(-4)*(10) = -40
(-5)*(8) = -40

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"18\"$ :

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First Number	Second Number	Sum
1	-40	1+(-40)=-39
2	-20	2+(-20)=-18
4	-10	4+(-10)=-6
5	-8	5+(-8)=-3
-1	40	-1+40=39
-2	20	-2+20=18
-4	10	-4+10=6
-5	8	-5+8=3

From the table, we can see that the two numbers $\"-2\"$ and $\"20\"$ add to $\"18\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"20\"$ both multiply to $\"-40\"$ and add to $\"18\"$

Now replace the middle term $\"18x\"$ with $\"-2x%2B20x\"$ . Remember, $\"-2\"$ and $\"20\"$ add to $\"18\"$ . So this shows us that $\"-2x%2B20x=18x\"$ .

$\"5x%5E2%2Bhighlight%28-2x%2B20x%29-8\"$ Replace the second term $\"18x\"$ with $\"-2x%2B20x\"$ .

$\"%285x%5E2-2x%29%2B%2820x-8%29\"$ Group the terms into two pairs.

$\"x%285x-2%29%2B%2820x-8%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%285x-2%29%2B4%285x-2%29\"$ Factor out $\"4\"$ 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.

$\"%28x%2B4%29%285x-2%29\"$ Combine like terms. Or factor out the common term $\"5x-2\"$

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Answer:

So $\"5%2Ax%5E2%2B18%2Ax-8\"$ factors to $\"%28x%2B4%29%285x-2%29\"$ .

In other words, $\"5%2Ax%5E2%2B18%2Ax-8=%28x%2B4%29%285x-2%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B4%29%285x-2%29\"$ to get $\"5%2Ax%5E2%2B18%2Ax-8\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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