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Question 1132875: solve the quadratic equations by factoring:
a) 4x^2=4x-1
b) 4n^2-15n-25=0
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
a) 
 to solve by factoring, you have to have everything on one side of the equation, set equal to 0 on the other side
The constant term is positive, and the middle term is negative. That means the two binomial factors will both have negative signs:
(??-??)(??-??)
There is only one way to get a constant term of 1; both of the constants in the binomial factors have to be 1:
(??-1)(??-1)
To get the 4x^2 leading term, there are two possibilities -- 2x and 2x for the leading terms of the binomials, or 4x and x. Trying the two options finds the right factorization:
(4x-1)(x-1) = 4x^2-5x+1 (wrong middle term)
(2x-1)(2x-1) = 4x^2-4x+1 (RIGHT!)
Note that this quadratic factors by one of the most common patterns, which you should learn to recognize easily: (a-b)(a-b) = a^2-2ab+b^2.
b) 4n^2-15n-25=0
There are many methods for factoring quadratics in which the leading term is not 1. I will show a couple.
To start with, the constant term this time is negative; that means the signs in the two binomial factors will be one positive and one negative.
(1) Logical trial and error...
I learned to do the factoring by simple trial and error. In this example, the leading terms in the binomial factors could again be either 4x and x, or 2x and 2x. The constant terms of the two binomials could be either 5 and 5, or 25 and 1.
25 and 1 for the constant terms is unlikely, so I would start by trying 5 and 5 for the constant terms of the binomials. With the opposite signs, I then have three possibilities; performing the multiplication finds the right one:
(2x-5)(2x+5) = 4x^2-25 (wrong middle term)
(4x-5)(x+5) = 4x^2+15x-25 (right size for the coefficient of the middle term, but wrong sign)
(4x+5)(x-5) = 4x^2-15x-25 (RIGHT!)
Note that it will always be the case that, if the signs of the two binomials are opposites and the factorization you try gives the right size but the wrong sign for the middle term, the right factorization will be with the signs in the two binomials switched.
(2) Another method....
Again the quadratic in this example is 4x^2-15x-25.
(1) multiply the leading coefficient and the constant: 4(-25) = -100
(2) Find two numbers whose product is -100 and whose sum is -15: they are -20 and 5
(3) Rewrite the quadratic as four terms, splitting the middle term as determined in step (2): 4x^2-20x+5x-25
(4) Factor the 4-term polynomial by grouping:
(4x^2-20x) + (5x-25) = 4x(x-5)+5(x-5) = (4x+5)(x-5)
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