Divide. Fraction 1: The quantity x squared plus 2 times x plus 1 over the quantity x squared minus 8 times x plus 16. Fraction 2: the quantity x plus 1 over the quantity x squared minus 16. Find Fraction 1 over Fraction 2.
A. The quantity x minus 1 times the quantity x plus 4, all over the quantity x plus 1
B. The quantity x plus 1 times the quantity x plus 4, all over the quantity x minus 4
C. The quantity x plus 1 times the quantity x plus 4, all over the quantity x minus 1
D. The quantity x minus 1 times the quantity x minus 4, all over the quantity x plus 4
To divide the two fractions, we need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of Fraction 2 is the quantity x squared minus 16 over the quantity x plus 1.
Therefore,
Fraction 1 ÷ Fraction 2 = (x^2 + 2x + 1)/(x^2 - 8x + 16) ÷ (x + 1)/(x^2 - 16)
= (x^2 + 2x + 1)/(x^2 - 8x + 16) * (x^2 - 16)/(x + 1)
= [(x + 1)(x + 1)]/[(x - 4)(x - 4)] * [(x - 4)(x + 4)]/(x + 1)
= (x + 1) * (x + 4) / (x - 4)
Therefore, the answer is (A) The quantity x minus 1 times the quantity x plus 4, all over the quantity x plus 1.
Simplify into one fraction.
start fraction 3 over x minus 3 end fraction minus start fraction 5 over x minus 2 end fraction
A. start fraction negative 2 x plus 9 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis
B. start fraction negative 2 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction
C. start fraction negative 2 left parenthesis x plus 9 right parenthesis over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction
D. start fraction 8 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction
To simplify the expression, we need to find a common denominator for the two fractions. The common denominator is (x - 3)(x - 2).
Therefore,
start fraction 3 over x minus 3 end fraction minus start fraction 5 over x minus 2 end fraction
= start fraction 3(x - 2)/[(x - 3)(x - 2)] - 5(x - 3)/[(x - 3)(x - 2)] end fraction
= start fraction (3x - 6 - 5x + 15)/[(x - 3)(x - 2)] end fraction
= start fraction (-2x + 9)/[(x - 3)(x - 2)] end fraction
Therefore, the answer is (A) start fraction negative 2 x plus 9 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis.
Solve.
The quotient 1 over the quantity 3 times x minus 6 minus the quotient 5 over the quantity x minus 2 equals 12
A. x equals 34 over 9
B. x equals negative 29 over 18
C. x equals negative 34 over 9
D. x equals 29 over 18
To solve the equation, we need to simplify the left side first.
Start with the first quotient:
start fraction 1 over 3x - 6 end fraction
We can factor out a 3 from the denominator:
start fraction 1 over 3(x - 2) end fraction
Now for the second quotient:
start fraction 5 over x - 2 end fraction
The LCD for this expression and 3(x - 2) is (x - 2)(3x - 6) = 3(x - 2)^2. We can rewrite the second quotient with this denominator:
start fraction 5(3x - 6) over 3(x - 2)^2 end fraction
Simplifying this expression:
start fraction 15x - 30 over 3(x - 2)^2 end fraction
Combining the two quotients:
start fraction 1 over 3(x - 2) end fraction - start fraction 15x - 30 over 3(x - 2)^2 end fraction
LCD for these two fractions is 3(x - 2)^2. Rewriting the expression with this denominator and simplifying:
start fraction (x - 2) - 5(3x - 6) end fraction / 3(x - 2)^2 = start fraction (-14x + 32)/3(x - 2)^2 end fraction
Now we can solve for x:
start fraction (-14x + 32)/3(x - 2)^2 = 12 end fraction
Multiplying both sides by 3(x - 2)^2:
-14x + 32 = 36(x - 2)^2
Expanding the right side:
-14x + 32 = 36(x^2 - 4x + 4)
Simplifying:
-14x + 32 = 36x^2 - 144x + 144
Rearranging and simplifying:
36x^2 - 130x + 112 = 0
We can solve this quadratic equation by factoring:
(9x - 28)(4x - 4) = 0
Solving for x, we get:
x = 28/9 or x = 1
However, we must check these solutions in the original equation to see if any of them produce denominators of zero.
For x = 1, the first quotient has a denominator of zero, which is not allowed.
For x = 28/9, the expression is valid:
start fraction 1 over 3(28/9) - 6 end fraction - start fraction 5 over 28/9 - 2 end fraction = 12
Simplifying and solving:
start fraction -1 over 25/3 end fraction = 12
-3/25 = 12
This is false, so there are no solutions to the equation.
Therefore, the answer is none of the above.
To solve this problem, we need to divide Fraction 1 by Fraction 2.
Fraction 1:
\(\frac{x^2 + 2x + 1}{x^2 - 8x + 16}\)
Fraction 2:
\(\frac{x + 1}{x^2 - 16}\)
To divide Fraction 1 by Fraction 2, we need to multiply Fraction 1 by the reciprocal of Fraction 2.
Reciprocal of Fraction 2:
\(\frac{x^2 - 16}{x + 1}\)
Now, let's multiply Fraction 1 by the reciprocal of Fraction 2:
\(\frac{x^2 + 2x + 1}{x^2 - 8x + 16} \times \frac{x^2 - 16}{x + 1}\)
Now, let's simplify the expression:
\(\frac{(x + 1)(x - 1)(x + 4)}{(x - 4)(x + 1)}\)
We can cancel out the common terms in the numerator and denominator:
\(\frac{(x - 1)(x + 4)}{(x - 4)}\)
Therefore, the answer is option D:
The quantity \(x - 1\) times the quantity \(x - 4\), all over the quantity \(x + 4\).
To find the division of two fractions, we need to multiply the first fraction by the reciprocal of the second fraction.
Given Fraction 1:
(x^2 + 2x + 1) / (x^2 - 8x + 16)
and Fraction 2:
(x + 1) / (x^2 - 16)
We can find Fraction 1 over Fraction 2 by multiplying Fraction 1 by the reciprocal of Fraction 2:
Fraction 1 / Fraction 2 = (x^2 + 2x + 1) / (x^2 - 8x + 16) * (x^2 - 16) / (x + 1)
Now, let's simplify the expression:
The numerator of Fraction 1 / Fraction 2 is:
(x^2 + 2x + 1) * (x^2 - 16)
Using the distributive property, we can expand this expression to:
x^4 - 16x^2 + 2x^3 - 32x + x^2 - 16
Combining like terms, we get:
x^4 + 2x^3 - 15x^2 - 32x - 16
The denominator of Fraction 1 / Fraction 2 is:
(x^2 - 8x + 16) * (x + 1)
Again, using the distributive property, we can expand this expression to:
x^3 + x^2 - 8x^2 - 8x + 16x + 16
Combining like terms, we get:
x^3 - 7x^2 + 8x + 16
Therefore, Fraction 1 / Fraction 2 can be written as:
(x^4 + 2x^3 - 15x^2 - 32x - 16) / (x^3 - 7x^2 + 8x + 16)
This expression cannot be simplified further, so option D. The quantity x minus 1 times the quantity x minus 4, all over the quantity x plus 4 is the correct answer.