Solve (p * r ^ 2 + q)/(2r - 1) + (p * r ^ 2 - q)/(2r + 1) + (4p * r ^ 3)/(1 - 4r ^ 2) 3 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { p r ^ { 2 } + q } { 2 r - 1 } + \frac { p r ^ { 2 } - q } { 2 r + 1 } + \frac { 4 p r ^ { 3 } } { 1 - 4 r ^ { 2 } }$$

Evaluate

$\frac{2q}{4r^{2}-1}$
$|r|\neq \frac{1}{2}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2r-1$ and $2r+1$ is $\left(2r-1\right)\left(2r+1\right)$. Multiply $\frac{pr^{2}+q}{2r-1}$ times $\frac{2r+1}{2r+1}$. Multiply $\frac{pr^{2}-q}{2r+1}$ times $\frac{2r-1}{2r-1}$.

$$\frac{\left(pr^{2}+q\right)\left(2r+1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Since $\frac{\left(pr^{2}+q\right)\left(2r+1\right)}{\left(2r-1\right)\left(2r+1\right)}$ and $\frac{\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(pr^{2}+q\right)\left(2r+1\right)+\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Do the multiplications in $\left(pr^{2}+q\right)\left(2r+1\right)+\left(pr^{2}-q\right)\left(2r-1\right)$.

$$\frac{2pr^{3}+pr^{2}+2qr+q+2pr^{3}-pr^{2}-2qr+q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Combine like terms in $2pr^{3}+pr^{2}+2qr+q+2pr^{3}-pr^{2}-2qr+q$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Factor $1-4r^{2}$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{\left(-2r-1\right)\left(2r-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(2r-1\right)\left(2r+1\right)$ and $\left(-2r-1\right)\left(2r-1\right)$ is $\left(2r-1\right)\left(2r+1\right)$. Multiply $\frac{4pr^{3}}{\left(-2r-1\right)\left(2r-1\right)}$ times $\frac{-1}{-1}$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$$

Since $\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}$ and $\frac{-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{4pr^{3}+2q-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$$

Combine like terms in $4pr^{3}+2q-4pr^{3}$.

$$\frac{2q}{\left(2r-1\right)\left(2r+1\right)}$$

Expand $\left(2r-1\right)\left(2r+1\right)$.

$$\frac{2q}{4r^{2}-1}$$

Expand

$\frac{2q}{4r^{2}-1}$
$|r|\neq \frac{1}{2}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2r-1$ and $2r+1$ is $\left(2r-1\right)\left(2r+1\right)$. Multiply $\frac{pr^{2}+q}{2r-1}$ times $\frac{2r+1}{2r+1}$. Multiply $\frac{pr^{2}-q}{2r+1}$ times $\frac{2r-1}{2r-1}$.

$$\frac{\left(pr^{2}+q\right)\left(2r+1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Since $\frac{\left(pr^{2}+q\right)\left(2r+1\right)}{\left(2r-1\right)\left(2r+1\right)}$ and $\frac{\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(pr^{2}+q\right)\left(2r+1\right)+\left(pr^{2}-q\right)\left(2r-1\right)}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Do the multiplications in $\left(pr^{2}+q\right)\left(2r+1\right)+\left(pr^{2}-q\right)\left(2r-1\right)$.

$$\frac{2pr^{3}+pr^{2}+2qr+q+2pr^{3}-pr^{2}-2qr+q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Combine like terms in $2pr^{3}+pr^{2}+2qr+q+2pr^{3}-pr^{2}-2qr+q$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{1-4r^{2}}$$

Factor $1-4r^{2}$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{4pr^{3}}{\left(-2r-1\right)\left(2r-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(2r-1\right)\left(2r+1\right)$ and $\left(-2r-1\right)\left(2r-1\right)$ is $\left(2r-1\right)\left(2r+1\right)$. Multiply $\frac{4pr^{3}}{\left(-2r-1\right)\left(2r-1\right)}$ times $\frac{-1}{-1}$.

$$\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}+\frac{-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$$

Since $\frac{4pr^{3}+2q}{\left(2r-1\right)\left(2r+1\right)}$ and $\frac{-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{4pr^{3}+2q-4pr^{3}}{\left(2r-1\right)\left(2r+1\right)}$$

Combine like terms in $4pr^{3}+2q-4pr^{3}$.

$$\frac{2q}{\left(2r-1\right)\left(2r+1\right)}$$

Expand $\left(2r-1\right)\left(2r+1\right)$.

$$\frac{2q}{4r^{2}-1}$$