Simplify the complex rational expression
3/(x+5)+ 6/(x+3) ÷ (3x+13)/(x^2+8x+15)
(please show steps i already know the answer i need to know the steps)
Use the fact that
(x^2+8x+15) = (x+5)(x+3)
Muliply numerator and denominator by that and you will get some big simplifications.
[3(x+3) + 6(x+5)] /(3x +13)
= (9x + 39)/(3x +13)
Do you see another simplication? Factor out "3" in the numerator.
To simplify the complex rational expression, you'll need to follow these steps:
Step 1: Combine the numerator of the main fraction.
Add the numerators of the two fractions in the numerator:
3/(x+5) + 6/(x+3) = (3(x+3) + 6(x+5))/(x+5)(x+3)
Simplify the numerator:
(3x + 9 + 6x + 30)/(x+5)(x+3) = (9x + 39)/(x+5)(x+3)
Step 2: Invert and multiply.
To divide by a fraction, you need to multiply by its reciprocal. In this case, we need to multiply by the reciprocal of the second fraction, (3x+13)/(x^2+8x+15).
Multiply the fractions:
(9x + 39)/(x+5)(x+3) * (x^2+8x+15)/(3x+13)
Step 3: Factor and simplify.
Factor the quadratic expression in the second fraction: x^2+8x+15 = (x+3)(x+5)
So now our expression is:
(9x + 39)/(x+5)(x+3) * (x+3)(x+5)/(3x+13)
Cancel out any common factors in the numerator and denominator:
(9x + 39)/(3x+13)
And that's the simplified complex rational expression: (9x + 39)/(3x+13).