Asked by Destinee Encina on Apr 26, 2024
Verified
Simplify the complex fraction. (x2−14x+45x2−10x+25) (16+6x−x2x2−3x−40) \frac { \left( \frac { x ^ { 2 } - 14 x + 45 } { x ^ { 2 } - 10 x + 25 } \right) } { \left( \frac { 16 + 6 x - x ^ { 2 } } { x ^ { 2 } - 3 x - 40 } \right) }(x2−3x−4016+6x−x2) (x2−10x+25x2−14x+45)
A) (x−9) (x+5) x−8,x≠−9,x≠5\frac { ( x - 9 ) ( x + 5 ) } { x - 8 } , x \neq - 9 , x \neq 5x−8(x−9) (x+5) ,x=−9,x=5
B) (5−x) (x+2) (x−8) (x+9) ,x≠−2,x≠9\frac { ( 5 - x ) ( x + 2 ) } { ( x - 8 ) ( x + 9 ) } , x \neq - 2 , x \neq 9(x−8) (x+9) (5−x) (x+2) ,x=−2,x=9
C) x+8x−5,x≠−8,x≠2\frac { x + 8 } { x - 5 } , x \neq - 8 , x \neq 2x−5x+8,x=−8,x=2
D) x−9x+2,x≠5,x≠9\frac { x - 9 } { x + 2 } , x \neq 5 , x \neq 9x+2x−9,x=5,x=9
E) (9−x) (x+5) (x−5) (x+2) ,x≠−5,x≠8\frac { ( 9 - x ) ( x + 5 ) } { ( x - 5 ) ( x + 2 ) } , x \neq - 5 , x \neq 8(x−5) (x+2) (9−x) (x+5) ,x=−5,x=8
Complex Fraction
A complex fraction is a fraction in which the numerator, the denominator, or both contain fractions themselves.
Simplify
The process of altering an expression to make it easier to work with, often by combining like terms or using mathematical properties.
- Learn the technique of making complex fractions simpler.
Verified Answer
YH
Yodany HamiltonApr 29, 2024
Final Answer :
E
Explanation :
First, we simplify the numerator and denominator separately.
Numerator:
x2−14x+45x2−10x+25\frac{x^2-14x+45}{x^2-10x+25}x2−10x+25x2−14x+45
(x−9)(x−5)(x−5)(x−5)\frac{(x-9)(x-5)}{(x-5)(x-5)}(x−5)(x−5)(x−9)(x−5)
(x−9)(x−5),x≠5\frac{(x-9)}{(x-5)}, x\neq 5(x−5)(x−9),x=5
Denominator:
16+6x−x2x2−3x−40\frac{16+6x-x^2}{x^2-3x-40}x2−3x−4016+6x−x2
−(x−9)(x+5)(x−8)(x+5)\frac{-(x-9)(x+5)}{(x-8)(x+5)}(x−8)(x+5)−(x−9)(x+5)
(9−x)(x−8),x≠−5\frac{(9-x)}{(x-8)}, x\neq -5(x−8)(9−x),x=−5
Now we can simplify the complex fraction:
(x2−14x+45x2−10x+25)(16+6x−x2x2−3x−40)\frac {\left(\frac{x^2-14x+45}{x^2-10x+25}\right)}{\left(\frac{16+6x-x^2}{x^2-3x-40}\right)}(x2−3x−4016+6x−x2)(x2−10x+25x2−14x+45)
(x−9)(x−5)(9−x)(x−8)\frac{\frac{(x-9)}{(x-5)}}{\frac{(9-x)}{(x-8)}}(x−8)(9−x)(x−5)(x−9)
(x−9)(x−8)(x−5)(9−x)\frac{(x-9)(x-8)}{(x-5)(9-x)}(x−5)(9−x)(x−9)(x−8)
(x−9)(x−8)−(x−5)(x−9)\frac{(x-9)(x-8)}{-(x-5)(x-9)}−(x−5)(x−9)(x−9)(x−8)
(x−8)−(x−5),x≠−5,x≠9\frac{(x-8)}{-(x-5)}, x\neq -5, x\neq 9−(x−5)(x−8),x=−5,x=9
Therefore, the answer is E: $\frac { ( 9 - x ) ( x + 5 ) } { ( x - 5 ) ( x + 2 ) } , x \neq - 5 , x \neq 8$.
Numerator:
x2−14x+45x2−10x+25\frac{x^2-14x+45}{x^2-10x+25}x2−10x+25x2−14x+45
(x−9)(x−5)(x−5)(x−5)\frac{(x-9)(x-5)}{(x-5)(x-5)}(x−5)(x−5)(x−9)(x−5)
(x−9)(x−5),x≠5\frac{(x-9)}{(x-5)}, x\neq 5(x−5)(x−9),x=5
Denominator:
16+6x−x2x2−3x−40\frac{16+6x-x^2}{x^2-3x-40}x2−3x−4016+6x−x2
−(x−9)(x+5)(x−8)(x+5)\frac{-(x-9)(x+5)}{(x-8)(x+5)}(x−8)(x+5)−(x−9)(x+5)
(9−x)(x−8),x≠−5\frac{(9-x)}{(x-8)}, x\neq -5(x−8)(9−x),x=−5
Now we can simplify the complex fraction:
(x2−14x+45x2−10x+25)(16+6x−x2x2−3x−40)\frac {\left(\frac{x^2-14x+45}{x^2-10x+25}\right)}{\left(\frac{16+6x-x^2}{x^2-3x-40}\right)}(x2−3x−4016+6x−x2)(x2−10x+25x2−14x+45)
(x−9)(x−5)(9−x)(x−8)\frac{\frac{(x-9)}{(x-5)}}{\frac{(9-x)}{(x-8)}}(x−8)(9−x)(x−5)(x−9)
(x−9)(x−8)(x−5)(9−x)\frac{(x-9)(x-8)}{(x-5)(9-x)}(x−5)(9−x)(x−9)(x−8)
(x−9)(x−8)−(x−5)(x−9)\frac{(x-9)(x-8)}{-(x-5)(x-9)}−(x−5)(x−9)(x−9)(x−8)
(x−8)−(x−5),x≠−5,x≠9\frac{(x-8)}{-(x-5)}, x\neq -5, x\neq 9−(x−5)(x−8),x=−5,x=9
Therefore, the answer is E: $\frac { ( 9 - x ) ( x + 5 ) } { ( x - 5 ) ( x + 2 ) } , x \neq - 5 , x \neq 8$.
Learning Objectives
- Learn the technique of making complex fractions simpler.