Can someone please help me with these two.
What is the quotient (6-x)/(x^2+2x-3)÷(x^2-4x-12)/(x^2+4x+3) in simplified form? State any restrictions on the variable.
Simplify the complex fraction
((x)/(x+2))/((1)/(x)+(1)/(x+2))
#1. The first fraction is
-(x-6)/(x-1)(x+3)
The other is
(x-6)(x+2)/(x+1)(x+3)
Dividing, you have
-(x-6)(x+1)(x+3) / (x-1)(x+3)(x-6)(x+2)
= -(x+1)/(x-1)(x+2)
x≠-3,-2,-1,1
#2.
x^2/(2(x+1))
What is the GCF of 18xy and 32xyz
Sure! Let's solve each problem step-by-step.
1. Quotient of two rational expressions:
Step 1: To divide one rational expression by another, we can multiply by the reciprocal of the second expression.
Step 2: Let's find the reciprocals of the expressions in both the numerator and denominator:
Reciprocal of (x^2+2x-3) is 1/(x^2+2x-3).
Reciprocal of (x^2-4x-12) is 1/(x^2-4x-12).
Step 3: Multiply the original expression by the reciprocal of the second expression:
(6-x)/(x^2+2x-3) * (x^2-4x-12)/(x^2+4x+3).
Step 4: Factorize the expressions in both the numerator and the denominator:
(x-6)/(x+3)(x-1) * (x+6)(x-2)/(x+1)(x+3).
Step 5: Cancel out common factors between the numerator and denominator:
(x-6)(x-2)/(x+1)(x-1).
Step 6: Simplify the expression:
(x^2 - 8x + 12)/(x^2 - 1).
Restrictions on the variable: The variable x cannot take the values that would make any of the denominators equal to zero. So, x cannot be equal to -3, 1, -1.
2. Simplifying the complex fraction:
Step 1: In a complex fraction, we can simplify it by multiplying the numerator and denominator by the least common denominator of all the fractions involved.
Step 2: In this case, the least common denominator of (x), (x+2), (1), and (x+2) is (x)(x+2).
Step 3: Multiply the numerator and denominator of the complex fraction by (x)(x+2):
((x)/(x+2))/((1)/(x)+(1)/(x+2)) * (x)(x+2)/(x)(x+2).
Step 4: Simplify the numerator and denominator:
(x(x))/(x+2) + (x(x+2))/(x) * (x)(x+2)/(x)(x+2).
Step 5: Simplify further:
x^2/(x+2) + (x(x+2))/(x).
Step 6: Combine the terms in the numerator:
(x^2 + x^2 + 2x)/(x+2).
Step 7: Simplify the expression:
(2x^2 + 2x)/(x+2).
And that's the simplified form of the complex fraction!
I hope this helps! Let me know if you have any further questions.
To find the quotient of the given rational expression (6-x)/(x^2+2x-3)÷(x^2-4x-12)/(x^2+4x+3) in simplified form, follow these steps:
Step 1: Simplify each rational expression separately.
First, factor the denominators:
x^2+2x-3 = (x-1)(x+3)
x^2-4x-12 = (x-6)(x+2)
x^2+4x+3 = (x+1)(x+3)
Now, rewrite the original expression:
(6-x)/(x^2+2x-3)÷(x^2-4x-12)/(x^2+4x+3)
= (6-x)/[(x-1)(x+3)] ÷ [(x-6)(x+2)]/[(x+1)(x+3)]
Step 2: Invert the second fraction and multiply:
= (6-x)/[(x-1)(x+3)] × [(x+1)(x+3)]/[(x-6)(x+2)]
Step 3: Cancel out common factors:
(x+3) can be cancelled out:
= (6-x)/[(x-1)] × [(x+1)]/[(x-6)(x+2)]
Step 4: Simplify further if possible:
Multiply the numerators together and the denominators together:
= (6-x)(x+1)/[(x-1)(x-6)(x+2)]
Step 5: Reorder the factors if necessary:
= (x+1)(6-x)/[(x-1)(x-6)(x+2)]
This is the simplified form of the quotient.
As for the restrictions on the variable, we should consider the denominators of the fractions involved. In this case, the restrictions are as follows:
(x-1)(x-6)(x+2) ≠ 0
This means that x cannot be equal to 1, 6, or -2.
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To simplify the complex fraction ((x)/(x+2))/((1)/(x)+(1)/(x+2)), follow these steps:
Step 1: Simplify each fraction separately.
The first fraction (x)/(x+2) doesn't require any further simplification.
The second fraction can be simplified by finding a common denominator:
(1)/(x) + (1)/(x+2) = (x+2)/(x(x+2)) + x/(x(x+2))
= (x+2 + x)/(x(x+2))
= (2x+2)/(x(x+2))
Step 2: Divide the two fractions by multiplying by the reciprocal of the second fraction:
((x)/(x+2))/((1)/(x)+(1)/(x+2)) = (x)/(x+2) * [(x(x+2))/(2x+2)]
= (x(x))/(x+2) * (x(x+2))/(2(x+1))
= (x^2 * x^2)/(2(x+1)(x+2))
Step 3: Simplify the expression further:
(x^2 * x^2) = x^4
2(x+1)(x+2) = 2(x^2 + 3x + 2) = 2x^2 + 6x + 4
Therefore, the simplified complex fraction is:
(x^4)/(2x^2 + 6x + 4)
No restrictions need to be considered for this particular complex fraction.