Question

X-5/3x+6 : (x-5/x+2)^2

Answer 1

X-5/3(x+2) : (x-5)^2/(x+2)^2
x-5/3(x+2) * (x+2)^2/(x-5)^2
1/3 * x+2/x-5
x+2/3x-15

Answer 2

To simplify the given expression, let's start by breaking it down step by step:

Expression: (x-5)/(3x+6) : (x-5/x+2)^2

Step 1: Simplify the fraction (x-5)/(3x+6)

The first step is to factor out the greatest common factor (GCF) from the numerator and the denominator:

(x-5)/(3x+6) = (x-5)/3(x+2)

Step 2: Simplify the expression in the denominator of the fraction: (x-5/x+2)^2

To square this expression, we need to square both the numerator and the denominator:

[(x-5)/(x+2)]^2 = [(x-5)^2]/[(x+2)^2]

Step 3: Simplify the squared expression

(x-5)^2 = x^2 - 10x + 25 (x+2)^2 = x^2 + 4x + 4

So, [(x-5)^2]/[(x+2)^2] = (x^2 - 10x + 25)/(x^2 + 4x + 4)

Step 4: Substitute the simplified expressions back into the original expression:

(x-5)/3(x+2) : (x^2 - 10x + 25)/(x^2 + 4x + 4)

Step 5: Invert and multiply (division by a fraction is the same as multiplying by its reciprocal):

(x-5)/3(x+2) * (x^2 + 4x + 4)/(x^2 - 10x + 25)

Step 6: Factor the numerator and denominator where possible:

Numerator: x^2 + 4x + 4 = (x+2)(x+2) = (x+2)^2 Denominator: x^2 - 10x + 25 = (x-5)(x-5) = (x-5)^2

So, the simplified expression is:

(x-5)/3(x+2) * (x+2)^2/(x-5)^2

Step 7: Cancel out common factors

Since (x-5) appears in both the numerator and the denominator, they can cancel out:

Final simplified expression:

1/3 * (x+2)/(x-5)