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Question 992537: How does this process change if the coefficient of the quadratic term is not either 1 or can be made to be 1 by removing the GCF? For example, consider a quadratic expression such as 14x2 + 29x - 15.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! equation is 14x^2 + 29x - 15 = 0
multiply the coefficient of the x^2 term by the constant term to get:
14 * 15 = 210
look for factors of 210 that will be equal to 210 when multiplied together and will be equal to 29 when added together.
since the constant term is negative, the two factors will have opposite signs because a positive times a negative is equal to a negative.
looking at all the factors of 210, i find that 35 and 6 are the factors that i need.
this is because 35 * 6 = 210 and 35 - 6 = 29.
the positive factor is 35.
the negative factor is 6.
now you split the middle term of the equation as shown below:
14x^2 + 29x - 15 = 0 becomes:
14x^2 + 35x - 6x - 15 = 0
now group the first two terms and the last two terms together to get:
(14x^2 + 35x) - (6x + 15) = 0
note the change of sign in the second of these grouped factors.
grouping with minus sings can be tricky.
- 6x - 15 becomes - (6x + 15) when you group those last two factors together.
that's because - (6x + 15) is equivalent to -6x - 15 when you expand it.
now factor out the common terms of reach group to get:
7x * (2x + 5) - 3 * (2x + 5)
what you are looking for here is that one of the factors in the secondf set is common to one of the factors in the first set.
the common factor that we were looking for here is 2x + 5
now you can factor out that common factor to get:
7x * (2x + 5) - 3 * (2x + 5) becomes:
(7x - 3) * (2x + 5)
those are your factors.
multiply them out and you will see that you get the original equation.
(7x - 3) * (2x + 5) =
7x * 2x = 14x^2
+ 7x * 5 = 35x
- 3 * 2x = -6x
- 3 * 5 = -15
combine like terms and you get:
14x^2 + 29x - 15.
that's your original equation so you're done.
this method is tricky but gets you to the right answer fairly quickly if you do it right.
same with the box method.
the other method is guess and check where you go through a ritual of trying dfifferent factors until you arrive at the right one.
if any of these methods prove difficult, then go to the quadratic formula.
that will be you the factors in all case, whether or not the quadratic equation is factorable or not, and whether or not the factors are real or not.
here's some references on factoring that you might find helpful.
http://www.purplemath.com/modules/solvquad.htm
http://www.purplemath.com/modules/factquad.htm
http://www.regentsprep.org/regents/math/algtrig/atv1/revfactorgrouping.htm
https://www.youtube.com/watch?v=Od38CRJNC5w
all kinds of stuff on the web.
all you need to do is a search on "factoring quadratic equations" or "factoring quadratics using box method" or "factoring quadratic equations using split the middle term method.
there are even other methods like the indian method.
personally, if i can't factor it easily using any of these methods, i go right to the quadratic formula since that is the method of last resort.
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