Question 1023568: 9x^2-65x-56
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with 9x^2 - 65x - 56
since the middle term is negative and the constant term is negative, you are looking for factors that will subtract from each other to get -65.
in general, it works this way.
if the x term is negative and the constant term is positive, then you are looking for factors that will add up to the middle term and be negative.
if the x term is positive and the constant term is positive, then you are looking for factors that will add up to the middle term and be positive.
if the x term is negative and the constant term is negative, then you are looking for factors that will subtract from each other and be negative.
if the x term is positive and the constant term is negative, then you are looking for factors that will subtract from each other and be positive.
up to now you have accomplished the following:
you started with 9x^2 - 65x - 56.
set that equal to 0 to get 9x^2 - 65x - 56 = 0
the equation is now in standard form and can be factored manually, if that's possible.
i use the split the middle term method, which is similar to the box method.
both methods are superior to the guess and check method, unless the factors are obvious from inspection.
if the quadratic equation is not able to be factored manually, then you would need to resort to the quadratic formula.
the quadratic formula will factor any quadratic equation, whether that equation has real roots or complex roots.
back to your equation.
the equation is now 9x^2 - 65x - 56 = 0
you would first try to take out the greatest common factor.
it does not appear that this equation has a greatest common factor other than 1.
it is not divisible by 9 and it is not divisible by 3.
multiply the coefficient of the x^2 term by the constant term to get 9 * 56 = 504.
that's the number you are going to work with.
now you need to take factors of this until you find factors that subtract from each other to get 65.
start with 1 * 504
next is 2 * 252
next is 4 * 126
next could be 8 * 63
you were looking for factors that are multiplied by each other to equal 504 and can be subtracted from each other to get 65.
1 * 504 had a difference too great.
so did 2 * 252 and 4 * 126.
8 * 63 has a difference that is too small, because 63 - 8 = 55.
if there are factors that are integers that subtract from each other to get 65, they have to be between 4 * 126 and 8 * 63.
i started backtracking and went to a factor of 7.
504 / 7 = 72.
so i got:
7 * 72 = 504
when i subtracted 7 from 72, i got 65.
that's the sect of factors i was looking for.
since the middle term is negative, then i need my factors to be + 7 and - 72.
here's where you split the middle term.
your equation is 9x^2 - 65x - 56 = 0
you split the middle term to get 9x^2 - 72x + 7x - 56 = 0
the middle term was - 65x and is now -72x + 7x which is equal to -65x once you add them together.
your equation is now 9x^2 - 72x + 7x - 56 = 0
you group the first two terms together and you group the last two terms together.
you get (9x^2 - 72x) + (7x - 56) = 0
you now want to factor the first set of terms and the second set of terms so that they'll both wind up with a common factor.
factor out a 9x from (9x^2 - 72x) to get 9x * (x - 8) = 0.
factor out a 7 from (7x - 56) to get 7 * (x - 8) = 0.
your equation now becomes 9x * (x - 8) + 7 * (x - 8) = 0.
your common factor is now (x - 8) which can be factored out.
your equation becomes:
(9x + 7) * (x - 8) = 0 after you factored out the common term of (x - 8).
these are the factors of your quadratic equation.
your solution is (9x + 7) * (x - 8) are factors of 9x^2 - 65x - 56.
if you need help in learning the various ways to factor a quadratic equation, try the following links.
http://www.purplemath.com/modules/quadform.htm
http://www.purplemath.com/modules/solvquad.htm
http://www.purplemath.com/modules/factquad.htm
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