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, write out the answer using numbers with out spelling things out
First, simplify the fractions:
Fraction 1: (x^2 - 16)/(2x^2 - 9x + 4)
Factor numerator and denominator:
((x + 4)(x - 4))/(2x - 1)(x - 4)
Simplify:
(x + 4)/(2x - 1)
Fraction 2: (2x^2 + 14x + 24)/(4x + 4)
Factor out 2 from numerator and denominator:
(2(x^2 + 7x + 12))/(4(x + 1))
Cancel out the 2s:
(x^2 + 7x + 12)/(2(x + 1))
Now we can divide Fraction 1 by Fraction 2:
[(x + 4)/(2x - 1)] / [(x^2 + 7x + 12)/(2(x + 1))]
When dividing fractions, we can flip the second fraction and multiply:
[(x + 4)/(2x - 1)] * [2(x + 1)/(x^2 + 7x + 12)]
Factor the quadratic in the denominator:
[(x + 4)/(2x - 1)] * [2(x + 1)/((x + 4)(x + 3))]
Cancel out the (x + 4) term in the numerator and denominator:
[(1)/(2x - 1)] * [2/(x + 3)]
Simplify:
2/(2x^2 + 5x - 3)
Therefore, Fraction 1 divided by Fraction 2 simplifies to 2/(2x^2 + 5x - 3).