SOLUTION: Simply the Rational Expression X^2-4X-12/6X^2-216 Could You please answer and explain it Thank You

Question 28506: Simply the Rational Expression
X^2-4X-12/6X^2-216
Could You please answer and explain it
Thank You
Answer by sdmmadam@yahoo.com(530) (Show Source): You can put this solution on YOUR website!
Simply the Rational Expression
X^2-4X-12/6X^2-216
Could You please answer and explain it
X^2-4X-12/6X^2-216
=[(X-6)(X+2)]/6[(X+6)(X-6)]
=(X+2)/6(X+6)
Answer:(X+2)/6(X+6)
Here is the Explanation:(in the form of a conversation!)
X^2-4X-12----(1)
is a quadratic expression in X. If it is factorisable, then the term in X should conform to getting expressed as the sum of two terms in X so that these two when multiplied should give the product of the square term and the constant term in the expression.
How do you split the term in X in the required way?
Multiply the square term and the constant term
(X^2)*(-12) = -(12X^2)
Find the factors of 12 that you see here in this product
(that is the numerical coefficient of X^2)
12= 1*2*2*3
Group the factors of 12 (taking into consideration all the factors) and form two parts in such a way that their sum is the coefficient of X.
That is (1*2*3) and (2) so that -4 = (-6+2) and (-6)*(+2) = -12
You may leave out the factor 1 as 1 multiplied by anything is the samething.
[if the term in X is negative and the product (of the square term and the constant term) is negative then to the larger of the two parts give negative sign and to the other the positive sign so that negative multiplied by positive is negative and larger negative added to smaller positive is negative]
So, here (-4x) = (-6x)+(2x) and (-6x)*(2x) = -12x^2
X^2-4X-12 becomes
=X^2+(-6X+2X)-12
Then group these four terms into a sum of two terms containing a common quantity.This common quantity will be a linear factor in X which we shall call p for convenience for the moment. Take p out. You get p multiplied by another linear factor in X. Put the value of p. Then you get the required answer namely the given expression as a product of two linear factors in X
=(X^2-6X)+(2x-12) by additive associativity
= X(X-6)+2(X-6)
= Xp+2p where p = (X-6)
= p(X+2)
=(X-6)(X+2)
6X^2-216----(2)
=6(X^2-36)
=6(X+6)(X-6) using the formula p^2-q^2 = (p+q)(p-q) where here p =X and q=6
The given problem is (1) divided by(2)
You observe that (X-6) is common and hence cancel it out

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