SOLUTION: Give exact and approximate solutions to three decimal places. y^2-10y+25=4 Thank you so much

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Question 258851: Give exact and approximate solutions to three decimal places.
y^2-10y+25=4


Thank you so much
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
y^2-10y+21=0
y=3 and y=7
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:



First NumberSecond NumberSum
1211+21=22
373+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).