SOLUTION: How to factor this trinomial ? 14uČ-33u-5

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Question 901358: How to factor this trinomial ?
14uČ-33u-5
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 14u%5E2-33u-5, we can see that the first coefficient is 14, the second coefficient is -33, and the last term is -5.



Now multiply the first coefficient 14 by the last term -5 to get %2814%29%28-5%29=-70.



Now the question is: what two whole numbers multiply to -70 (the previous product) and add to the second coefficient -33?



To find these two numbers, we need to list all of the factors of -70 (the previous product).



Factors of -70:

1,2,5,7,10,14,35,70

-1,-2,-5,-7,-10,-14,-35,-70



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



These factors pair up and multiply to -70.

1*(-70) = -70
2*(-35) = -70
5*(-14) = -70
7*(-10) = -70
(-1)*(70) = -70
(-2)*(35) = -70
(-5)*(14) = -70
(-7)*(10) = -70


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -33:



First NumberSecond NumberSum
1-701+(-70)=-69
2-352+(-35)=-33
5-145+(-14)=-9
7-107+(-10)=-3
-170-1+70=69
-235-2+35=33
-514-5+14=9
-710-7+10=3




From the table, we can see that the two numbers 2 and -35 add to -33 (the middle coefficient).



So the two numbers 2 and -35 both multiply to -70 and add to -33



Now replace the middle term -33u with 2u-35u. Remember, 2 and -35 add to -33. So this shows us that 2u-35u=-33u.



14u%5E2%2Bhighlight%282u-35u%29-5 Replace the second term -33u with 2u-35u.



%2814u%5E2%2B2u%29%2B%28-35u-5%29 Group the terms into two pairs.



2u%287u%2B1%29%2B%28-35u-5%29 Factor out the GCF 2u from the first group.



2u%287u%2B1%29-5%287u%2B1%29 Factor out 5 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.



%282u-5%29%287u%2B1%29 Combine like terms. Or factor out the common term 7u%2B1



===============================================================



Answer:



So 14%2Au%5E2-33%2Au-5 factors to %282u-5%29%287u%2B1%29.



In other words, 14%2Au%5E2-33%2Au-5=%282u-5%29%287u%2B1%29.



Note: you can check the answer by expanding %282u-5%29%287u%2B1%29 to get 14%2Au%5E2-33%2Au-5 or by graphing the original expression and the answer (the two graphs should be identical).



Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for u:


Starting with the general quadratic


au%5E2%2Bbu%2Bc=0


the general solution using the quadratic equation is:


u+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 14%2Au%5E2-33%2Au-5=0 ( notice a=14, b=-33, and c=-5)





u+=+%28--33+%2B-+sqrt%28+%28-33%29%5E2-4%2A14%2A-5+%29%29%2F%282%2A14%29 Plug in a=14, b=-33, and c=-5




u+=+%2833+%2B-+sqrt%28+%28-33%29%5E2-4%2A14%2A-5+%29%29%2F%282%2A14%29 Negate -33 to get 33




u+=+%2833+%2B-+sqrt%28+1089-4%2A14%2A-5+%29%29%2F%282%2A14%29 Square -33 to get 1089 (note: remember when you square -33, you must square the negative as well. This is because %28-33%29%5E2=-33%2A-33=1089.)




u+=+%2833+%2B-+sqrt%28+1089%2B280+%29%29%2F%282%2A14%29 Multiply -4%2A-5%2A14 to get 280




u+=+%2833+%2B-+sqrt%28+1369+%29%29%2F%282%2A14%29 Combine like terms in the radicand (everything under the square root)




u+=+%2833+%2B-+37%29%2F%282%2A14%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




u+=+%2833+%2B-+37%29%2F28 Multiply 2 and 14 to get 28


So now the expression breaks down into two parts


u+=+%2833+%2B+37%29%2F28 or u+=+%2833+-+37%29%2F28


Lets look at the first part:


x=%2833+%2B+37%29%2F28


u=70%2F28 Add the terms in the numerator

u=5%2F2 Divide


So one answer is

u=5%2F2




Now lets look at the second part:


x=%2833+-+37%29%2F28


u=-4%2F28 Subtract the terms in the numerator

u=-1%2F7 Divide


So another answer is

u=-1%2F7


So our solutions are:

u=5%2F2 or u=-1%2F7