document.write( "Question 258851: Give exact and approximate solutions to three decimal places.\r
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Algebra.Com's Answer #190572 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
y^2-10y+21=0
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"y%5E2-10y%2B21\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-10\"$ , and the last term is $\"21\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"21\"$ to get $\"%281%29%2821%29=21\"$ .

Now the question is: what two whole numbers multiply to $\"21\"$ (the previous product) and add to the second coefficient $\"-10\"$ ?

To find these two numbers, we need to list all of the factors of $\"21\"$ (the previous product).

Factors of $\"21\"$ :

1,3,7,21

-1,-3,-7,-21

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

These factors pair up and multiply to $\"21\"$ .

1*21 = 21
3*7 = 21
(-1)*(-21) = 21
(-3)*(-7) = 21

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-10\"$ :

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First Number	Second Number	Sum
1	21	1+21=22
3	7	3+7=10
-1	-21	-1+(-21)=-22
-3	-7	-3+(-7)=-10

From the table, we can see that the two numbers $\"-3\"$ and $\"-7\"$ add to $\"-10\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"-7\"$ both multiply to $\"21\"$ and add to $\"-10\"$

Now replace the middle term $\"-10y\"$ with $\"-3y-7y\"$ . Remember, $\"-3\"$ and $\"-7\"$ add to $\"-10\"$ . So this shows us that $\"-3y-7y=-10y\"$ .

$\"y%5E2%2Bhighlight%28-3y-7y%29%2B21\"$ Replace the second term $\"-10y\"$ with $\"-3y-7y\"$ .

$\"%28y%5E2-3y%29%2B%28-7y%2B21%29\"$ Group the terms into two pairs.

$\"y%28y-3%29%2B%28-7y%2B21%29\"$ Factor out the GCF $\"y\"$ from the first group.

$\"y%28y-3%29-7%28y-3%29\"$ Factor out $\"7\"$ 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.

$\"%28y-7%29%28y-3%29\"$ Combine like terms. Or factor out the common term $\"y-3\"$

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Answer:

So $\"y%5E2-10%2Ay%2B21\"$ factors to $\"%28y-7%29%28y-3%29\"$ .

In other words, $\"y%5E2-10%2Ay%2B21=%28y-7%29%28y-3%29\"$ .

Note: you can check the answer by expanding $\"%28y-7%29%28y-3%29\"$ to get $\"y%5E2-10%2Ay%2B21\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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