t Fraction 1: the quantity x squared minus 16 over the quantity 2 times x squared minus 9 times x plus 4; Fraction 2: the quantity 2 times x squared plus 14 times x plus 24 over the quantity 4 times x plus 4; Find Fraction 1 divided by Fraction 2.
Find the quotient
To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. In this case:
Fraction 1 divided by Fraction 2 = Fraction 1 * (Fraction 2)^(-1)
We need to find the reciprocal of Fraction 2 and then multiply it by Fraction 1. Let's start by finding the reciprocal of Fraction 2:
(Fraction 2)^(-1) = (4x+4)/(2x^2+14x+24)
Now we can multiply Fraction 1 by this reciprocal:
Fraction 1 * (Fraction 2)^(-1) = [(x^2-16)/(2x^2-9x+4)] * [(4x+4)/(2x^2+14x+24)]
To simplify this expression, we can factor the numerators and denominators of both fractions:
Fraction 1 * (Fraction 2)^(-1) = [(x+4)(x-4)/((2x-1)(x-4))] * [4(x+1)/(2(x+3)(x+2))]
We can cancel out the common factors of (x+4) and (x+1):
Fraction 1 * (Fraction 2)^(-1) = [(x-4)/((2x-1)(x-4))] * [4/(2(x+3)(x+2))]
We can simplify the fraction by canceling out (x-4) and simplifying the denominator:
Fraction 1 * (Fraction 2)^(-1) = [1/((2x-1)(x+3)(x+2))]
Therefore, Fraction 1 divided by Fraction 2 is equal to 1 over the quantity 2 times x minus 1 times x plus 3 times x plus 2.