Question

What is (x+5)/4x · 12x²/x²+7x+10 in its lowest terms?

Answer 1

3

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2

x+2

3x

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Step-by-step explanation:

To simplify the expression \(\frac{(x+5)}{4x} \cdot \frac{12x^2}{x^2+7x+10}\) and express it in its lowest terms, follow these steps:

1. Factor the quadratic expression in the denominator of the second fraction (\(x^2 + 7x + 10\)):

\[x^2 + 7x + 10 = (x + 5)(x + 2)\]

2. Rewrite the expression with the factored form:

\[\frac{(x+5)}{4x} \cdot \frac{12x^2}{(x + 5)(x + 2)}\]

3. Simplify by canceling common factors:

\[\frac{\cancel{(x+5)}}{\cancel{4x}} \cdot \frac{3 \cdot \cancel{4} \cdot x \cdot \cancel{x}}{\cancel{(x + 5)}(x + 2)}\]

4. Multiply the remaining factors:

\[\frac{3x}{x + 2}\]

So, \(\frac{(x+5)}{4x} \cdot \frac{12x^2}{x^2+7x+10}\) simplifies to \(\frac{3x}{x + 2}\) in its lowest terms.

Answer 2

To simplify the expression (x+5)/4x · 12x²/x²+7x+10 in its lowest terms, you can simplify each part separately and then combine them. The simplified expression is (3x+7)/(x+2).

Step-by-step explanation:

To simplify the expression (x+5)/4x · 12x²/x²+7x+10 in its lowest terms, we need to simplify each part separately and then combine them. Let's break it down:

1. Simplify (x+5)/4x. This can be simplified further by canceling out the x terms: (x+5)/(4x) = (x/4x)+(5/4x) = 1/4 + 5/4x = (1+5/x)/4.
2. Simplify 12x²/x²+7x+10. This can be factored: (12x²)/((x+5)(x+2)). We can then see that the x+5 term in the numerator cancels with the x+5 term in the denominator, leaving us with 12x²/(x+2).
3. Combining the simplified expressions: ((1+5/x)/4) · (12x²/(x+2)) = (12x+30)/(4(x+2)) = (6x+15)/(2(x+2)) = (3x+7)/(x+2).

Therefore, (x+5)/4x · 12x²/x²+7x+10, in its lowest terms, is (3x+7)/(x+2).