How do you find the quotient of #(5x^2)/(x^2-5x+4)div(10x)/(x-1)#?

Question

Isaiah Anthony · Accepted Answer

See a solution process below:

First, factor the denominator of the fraction on the left: #(5x^2)/(x^2 - 5x + 4) -: (10x)/(x - 1) => (5x^2)/((x - 4)(x - 1)) -: (10x)/(x - 1)# Next, rewrite this expression as: #((5x^2)/((x - 4)(x - 1)))/((10x)/(x - 1))# Then, use this rule of dividing fractions to rewrite the expression again and find the quotient: #(color(red)((5x^2))/color(blue)(((x - 4)(x - 1))))/(color(green)((10x))/color(purple)((x - 1))) => (color(red)((5x^2)) xx color(purple)((x - 1)))/(color(blue)(((x - 4)(x - 1))) xx color(green)((10x)))# #(color(red)((cancel(5)cancel(x^2)x)) xx cancel(color(purple)((x - 1))))/(color(blue)(((x - 4)cancel((x - 1)))) xx color(green)((cancel(10)2cancel(x)))) =># #x/(2(x - 4))# or #x/(2x - 8)#

Jace Anderson · Answer

undefined To find the quotient of (5x^2)/(x^2-5x+4) divided by (10x)/(x-1), we can simplify the expression by multiplying the numerator and denominator of the first fraction by the reciprocal of the second fraction.

First, let's find the reciprocal of (10x)/(x-1), which is (x-1)/(10x).

Now, we can rewrite the expression as (5x^2)/(x^2-5x+4) multiplied by (x-1)/(10x).

Next, we can simplify the expression by canceling out common factors.

The x in the numerator and denominator can be canceled out, leaving us with (5x)/(x^2-5x+4) multiplied by (x-1)/(10).

Finally, we can multiply the numerators and denominators together to find the quotient.

The quotient is (5x(x-1))/(10(x^2-5x+4)).

Nathan Axel · Answer

undefined To find the quotient of $ \frac{5x^2}{x^2-5x+4} \div \frac{10x}{x-1} $, you can follow these steps:

1. Factor both the numerator and the denominator of each fraction, if possible.
2. Rewrite the division as multiplication by the reciprocal of the second fraction.
3. Simplify by canceling out common factors in the numerator and denominator.
4. Perform any necessary simplification or further factorization.
5. Write the final expression in simplest form.

Let's proceed with the solution:

1. Factor the denominators:
$$ x^2 - 5x + 4 = (x - 4)(x - 1) $$
$$ x - 1 \text{ (already factored)} $$

2. Rewrite the division as multiplication by the reciprocal:
$$ \frac{5x^2}{x^2-5x+4} \times \frac{x-1}{10x} $$

3. Cancel out common factors:
$$ \frac{5x}{x-4} \times \frac{x-1}{10x} $$

4. Further simplification:
$$ \frac{5}{2(x-4)} \times \frac{x-1}{x} $$

5. Write the final expression in simplest form:
$$ \frac{5(x-1)}{2x(x-4)} $$

Therefore, the quotient of $ \frac{5x^2}{x^2-5x+4} \div \frac{10x}{x-1} $ simplifies to $ \frac{5(x-1)}{2x(x-4)} $.