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Question 414515: Choose the simplified expression. : (x)/(x^2-4)+ (7)/(3x+6)
also
Choose the simplified expression. : (m)/(m+3)-(6m)/(m^2-9)
Answer by woolybear(13)   (Show Source):
You can put this solution on YOUR website! (x)/(x^(2)-4)+(7)/(3x+6)
The binomial can be factored using the difference of squares formula, because both terms are perfect squares. The difference of squares formula is a^(2)-b^(2)=(a-b)(a+b).
(x)/((x-2)(x+2))+(7)/(3x+6)
Factor out the GCF of 3 from each term in the polynomial.
(x)/((x-2)(x+2))+(7)/(3(x)+3(2))
Factor out the GCF of 3 from 3x+6.
(x)/((x-2)(x+2))+(7)/(3(x+2))
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 3(x+2)(x-2). The (x)/((x-2)(x+2)) expression needs to be multiplied by ((3))/((3)) to make the denominator 3(x+2)(x-2). The (7)/(3(x+2)) expression needs to be multiplied by ((x-2))/((x-2)) to make the denominator 3(x+2)(x-2).
(x)/((x-2)(x+2))*(3)/(3)+(7)/(3(x+2))*(x-2)/(x-2)
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 3(x+2)(x-2).
(x(3))/(3(x+2)(x-2))+(7)/(3(x+2))*(x-2)/(x-2)
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 3(x+2)(x-2).
(x(3))/(3(x+2)(x-2))+(7(x-2))/(3(x+2)(x-2))
The numerators of expressions that have equal denominators can be combined. In this case, (x(3))/(3(x+2)(x-2)) and ((7(x-2)))/(3(x+2)(x-2)) have the same denominator of 3(x+2)(x-2), so the numerators can be combined.
(x(3)+(7(x-2)))/(3(x+2)(x-2))
Simplify the numerator of the expression.
(3x+7x-14)/(3(x+2)(x-2))
Since 3x and 7x are like terms, add 7x to 3x to get 10x.
(10x-14)/(3(x+2)(x-2))
Factor out the GCF of 2 from each term in the polynomial.
(2(5x)+2(-7))/(3(x+2)(x-2))
Factor out the GCF of 2 from 10x-14.
(2(5x-7))/(3(x+2)(x-2))
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