What is the quotient 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. ?
A. 1 over the quantity x plus 3
B. The quantity 2 times x plus 2 over the quantity x plus 3
C. The quantity 2 times x plus 2 over the quantity 2 times x squared plus 5 times x minus 3
D. 2 times the quantity x plus 1 over the quantity 2 times x squared minus 5 times x minus 3
Lyle
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Given
Fraction 1: (x^2-16)/(2x^2-9x+4)
Fraction 2: (2x^2+14x+24)/(4x+4)
Fraction 1 divided by Fraction 2:
(x^2-16)/(2x^2-9x+4) ÷ (2x^2+14x+24) ÷ (4x+4)
(x^2-16)/(2x^2-9x+4) (2x^2+14x+24) ÷ (4x+4) (4x+4)
(2x^2-x^2+2x^2-9x+16)/(3x^2+2x^2-9x+36) (-2x+14) ÷ (-2x+2)
5x-3/6x^2+5x-39
Multiply both the numerator and denominator by 3:
(15x-9)/(18x^2+9x-117)
Divide both the numerator and denominator by x+3:
(15x-9)/(18x^2+9x-117) ÷ x+3
(15-9) ÷ (18(x^2-9)) ÷ (x+3)
6 ÷ (-9x+36) ÷ (x+3)
6 ÷ (-9x+36+x-3)
= 6 ÷ 33 = 2/11
Therefore, Fraction 1 divided by Fraction 2 is equal to 2/11 or 0.181818 repeating.
Option B, The quantity 2 times x plus 2 over the quantity x plus 3 is incorrect because it has both numerator and denominator in terms of x only
Option C, The quantity 2 times x plus 2 over the quantity 2 times x squared plus 5 times x minus 3 is incorrect because it has a squared variable.
Option D, 2 times the quantity x plus 1 over the quantity 2 times x squared minus 5 times x minus 3 is incorrect because it has a squared variable.
What is the quotient 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. ?
A. 1 over the quantity x plus 3
B. The quantity 2 times x plus 2 over the quantity x plus 3
C. The quantity 2 times x plus 2 over the quantity 2 times x squared plus 5 times x minus 3
D. 2 times the quantity x plus 1 over the quantity 2 times x squared minus 5 times x minus 3
The correct answer is B. The quantity 2 times x plus 2 over the quantity x plus 3.
To get to this answer, we first need to simplify both fractions:
Fraction 1: (x^2-16)/(2x^2-9x+4) can be factored as (x+4)(x-4)/(2x-1)(x-4)
Fraction 2: (2x^2+14x+24)/(4x+4) can be simplified as 2(x^2+7x+12)/4(x+1)
We can now rewrite the division of Fraction 1 by Fraction 2 as the product of Fraction 1 by the reciprocal of Fraction 2:
((x+4)(x-4)/(2x-1)(x-4)) / (2(x^2+7x+12)/4(x+1))
Simplifying the expression by cancelling out (x-4) and then simplifying common terms:
((x+4)/(2x-1)) * ((x+1)/ (x^2+7x+12)) = (x+4)(x+1) / (2x-1)(x+3)
This expression can be further simplified as (x+4)(x+1)/(2x-1)(x+3) = (2x+2)/(x+3) = 2(x+1)/(x+3), which matches answer B.
To find the quotient of two fractions, you need to divide the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second fraction.
Let's evaluate the numerator and denominator of Fraction 1 divided by Fraction 2 separately.
Numerator:
The numerator of Fraction 1 is given as (x² - 16).
The numerator of Fraction 2 is given as (2x² + 14x + 24).
To divide these two numerators, we need to perform the following calculation: (x² - 16) / (2x² + 14x + 24).
Denominator:
The denominator of Fraction 1 is given as (2x² - 9x + 4).
The denominator of Fraction 2 is given as (4x + 4).
To divide these two denominators, we need to perform the following calculation: (2x² - 9x + 4) / (4x + 4).
Now that we have the numerator and denominator of the quotient, we can write the entire expression as Fraction 1 divided by Fraction 2:
(x² - 16) / (2x² + 14x + 24) divided by (2x² - 9x + 4) / (4x + 4).
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction. In this case, we need to multiply the numerator by the reciprocal of the denominator:
(x² - 16) / (2x² + 14x + 24) * (4x + 4) / (2x² - 9x + 4).
Simplifying this expression will give us the final answer.