Math / Algebra

Question $\frac{x+4}{2 x-2} \div \frac{2 x^{2}+13 x+20}{4 x^{2}-4 x}$ answer as a single fraction in its simplest form

Studdy Solution

STEP 1

What is this asking? We're asked to divide one algebraic fraction by another and simplify the result. Watch out! Don't forget to flip the second fraction before multiplying, and always factor everything you can before simplifying!

STEP 2

1. Factor the numerators and denominators.
2. Flip and multiply.
3. Simplify.

STEP 3

$x+4$ is already factored.
Nothing more to do here!

STEP 4

We can factor out a $2$ from $2x-2$ to get $2(x-1)$ .

STEP 5

We're looking for two numbers that multiply to $(2 \cdot 20) = 40$ and add up to $13$ .
Those numbers are $8$ and $5$ .
So, we can rewrite $2x^2 + 13x + 20$ as $2x^2 + 8x + 5x + 20$ .
Factoring by grouping, we get $2x(x+4) + 5(x+4)$ , which simplifies to $(2x+5)(x+4)$ .

STEP 6

We can factor out a $4x$ from $4x^2 - 4x$ to get $4x(x-1)$ .

STEP 7

Remember, dividing by a fraction is the same as multiplying by its reciprocal.
So, we flip the second fraction and change the division to multiplication: $\frac{x+4}{2(x-1)} \cdot \frac{4x(x-1)}{(2x+5)(x+4)}$

STEP 8

We can divide to one the $(x+4)$ terms in the numerator and denominator.
We can also divide to one the $(x-1)$ terms.
Don't forget we can simplify $4x$ and $2$ as well, by dividing both by $2$ , leaving us with $2x$ and $1$ respectively. $\frac{1}{1} \cdot \frac{2x}{2x+5}$

STEP 9

Now, we multiply the numerators together and the denominators together: $\frac{2x}{2x+5}$

STEP 10

Our simplified answer is $\frac{2x}{2x+5}$ .

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Question $\frac{x+4}{2 x-2} \div \frac{2 x^{2}+13 x+20}{4 x^{2}-4 x}$ answer as a single fraction in its simplest form

Studdy Solution

STEP 1

What is this asking? We're asked to divide one algebraic fraction by another and simplify the result. Watch out! Don't forget to flip the second fraction before multiplying, and always factor everything you can before simplifying!

STEP 2

1. Factor the numerators and denominators.
2. Flip and multiply.
3. Simplify.

STEP 3

$x+4$ is already factored.
Nothing more to do here!

STEP 4

We can factor out a $2$ from $2x-2$ to get $2(x-1)$ .

STEP 5

We're looking for two numbers that multiply to $(2 \cdot 20) = 40$ and add up to $13$ .
Those numbers are $8$ and $5$ .
So, we can rewrite $2x^2 + 13x + 20$ as $2x^2 + 8x + 5x + 20$ .
Factoring by grouping, we get $2x(x+4) + 5(x+4)$ , which simplifies to $(2x+5)(x+4)$ .

STEP 6

We can factor out a $4x$ from $4x^2 - 4x$ to get $4x(x-1)$ .

STEP 7

Remember, dividing by a fraction is the same as multiplying by its reciprocal.
So, we flip the second fraction and change the division to multiplication: $\frac{x+4}{2(x-1)} \cdot \frac{4x(x-1)}{(2x+5)(x+4)}$

STEP 8

We can divide to one the $(x+4)$ terms in the numerator and denominator.
We can also divide to one the $(x-1)$ terms.
Don't forget we can simplify $4x$ and $2$ as well, by dividing both by $2$ , leaving us with $2x$ and $1$ respectively. $\frac{1}{1} \cdot \frac{2x}{2x+5}$

STEP 9

Now, we multiply the numerators together and the denominators together: $\frac{2x}{2x+5}$

STEP 10

Our simplified answer is $\frac{2x}{2x+5}$ .

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Questionx+42x−2÷2x2+13x+204x2−4x\frac{x+4}{2 x-2} \div \frac{2 x^{2}+13 x+20}{4 x^{2}-4 x}2x−2x+4​÷4x2−4x2x2+13x+20​ answer as a single fraction in its simplest form

Studdy Solution

STEP 1

What is this asking? We're asked to divide one algebraic fraction by another and simplify the result. Watch out! Don't forget to flip the second fraction before multiplying, and always factor everything you can before simplifying!

STEP 2

1. Factor the numerators and denominators.2. Flip and multiply.3. Simplify.

STEP 3

x+4x+4x+4 is already factored.Nothing more to do here!

STEP 4

We can **factor out** a 222 from 2x−22x-22x−2 to get 2(x−1)2(x-1)2(x−1).

STEP 5

STEP 6

We can **factor out** a 4x4x4x from 4x2−4x4x^2 - 4x4x2−4x to get 4x(x−1)4x(x-1)4x(x−1).

STEP 7

Remember, dividing by a fraction is the same as multiplying by its reciprocal.So, we flip the second fraction and change the division to multiplication: x+42(x−1)⋅4x(x−1)(2x+5)(x+4) \frac{x+4}{2(x-1)} \cdot \frac{4x(x-1)}{(2x+5)(x+4)} 2(x−1)x+4​⋅(2x+5)(x+4)4x(x−1)​

STEP 8

STEP 9

Now, we multiply the numerators together and the denominators together: 2x2x+5 \frac{2x}{2x+5} 2x+52x​

STEP 10

Our simplified answer is 2x2x+5\frac{2x}{2x+5}2x+52x​.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question $\frac{x+4}{2 x-2} \div \frac{2 x^{2}+13 x+20}{4 x^{2}-4 x}$ answer as a single fraction in its simplest form

1. Factor the numerators and denominators.
2. Flip and multiply.
3. Simplify.

$x+4$ is already factored.
Nothing more to do here!

We can factor out a $2$ from $2x-2$ to get $2(x-1)$ .

We can factor out a $4x$ from $4x^2 - 4x$ to get $4x(x-1)$ .

Remember, dividing by a fraction is the same as multiplying by its reciprocal.
So, we flip the second fraction and change the division to multiplication: $\frac{x+4}{2(x-1)} \cdot \frac{4x(x-1)}{(2x+5)(x+4)}$

Now, we multiply the numerators together and the denominators together: $\frac{2x}{2x+5}$

Our simplified answer is $\frac{2x}{2x+5}$ .