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QuestionSimplify the expression: 15x+362x+2+3x+3\frac{15 x+36}{\frac{2}{x+2}+\frac{3}{x+3}}?

Studdy Solution

STEP 1

Assumptions1. We are given the expression 15x+36x++3x+3\frac{15 x+36}{\frac{}{x+}+\frac{3}{x+3}} . We are assuming that xx is a real number and x≠−,−3x \neq -, -3 to avoid division by zero in the denominators.

STEP 2

The first step is to simplify the denominator. We can do this by finding a common denominator for the two fractions in the denominator.
2x+2+x+=2(x+)+(x+2)(x+2)(x+)\frac{2}{x+2}+\frac{}{x+} = \frac{2(x+) +(x+2)}{(x+2)(x+)}

STEP 3

Now, simplify the numerator of the new fraction in the denominator.
2(x+3)+3(x+2)(x+2)(x+3)=2x+6+3x+6(x+2)(x+3)\frac{2(x+3) +3(x+2)}{(x+2)(x+3)} = \frac{2x+6 +3x+6}{(x+2)(x+3)}

STEP 4

Combine like terms in the numerator of the new fraction in the denominator.
2x+6+3x+6(x+2)(x+3)=x+12(x+2)(x+3)\frac{2x+6 +3x+6}{(x+2)(x+3)} = \frac{x+12}{(x+2)(x+3)}

STEP 5

Now, substitute the simplified denominator back into the original expression.
15x+365x+12(x+2)(x+3)\frac{15 x+36}{\frac{5x+12}{(x+2)(x+3)}}

STEP 6

To simplify this complex fraction, we can multiply the numerator and the denominator by the denominator of the fraction in the denominator.
(15x+36)(x+2)(x+3)5x+12\frac{(15 x+36)(x+2)(x+3)}{5x+12}

STEP 7

Now, distribute (x+2)(x+3)(x+2)(x+3) in the numerator.
(15x+36)(x+2)(x+3)=(15x2+66x+108)(15 x+36)(x+2)(x+3) = (15x^2 +66x +108)

STEP 8

Finally, substitute the simplified numerator back into the expression.
15x2+66x+1085x+12\frac{15x^2 +66x +108}{5x+12}This is the simplified form of the given expression.

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