Solve (a + 12)/(4a + 16) - (a + 4)/(4a - 16) + 19/(a ^ 2 - 16) | 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

$$\frac{a+12}{4a+16}-\frac{a+4}{4a-16}+\frac{19}{a^{2}-16}$$

Evaluate

$\frac{3}{a^{2}-16}$

Short Solution Steps

Factor $4a+16$. Factor $4a-16$.

$$\frac{a+12}{4\left(a+4\right)}-\frac{a+4}{4\left(a-4\right)}+\frac{19}{a^{2}-16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(a+4\right)$ and $4\left(a-4\right)$ is $4\left(a-4\right)\left(a+4\right)$. Multiply $\frac{a+12}{4\left(a+4\right)}$ times $\frac{a-4}{a-4}$. Multiply $\frac{a+4}{4\left(a-4\right)}$ times $\frac{a+4}{a+4}$.

$$\frac{\left(a+12\right)\left(a-4\right)}{4\left(a-4\right)\left(a+4\right)}-\frac{\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Since $\frac{\left(a+12\right)\left(a-4\right)}{4\left(a-4\right)\left(a+4\right)}$ and $\frac{\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+12\right)\left(a-4\right)-\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Do the multiplications in $\left(a+12\right)\left(a-4\right)-\left(a+4\right)\left(a+4\right)$.

$$\frac{a^{2}-4a+12a-48-a^{2}-4a-4a-16}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Combine like terms in $a^{2}-4a+12a-48-a^{2}-4a-4a-16$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Factor $a^{2}-16$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{\left(a-4\right)\left(a+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(a-4\right)\left(a+4\right)$ and $\left(a-4\right)\left(a+4\right)$ is $4\left(a-4\right)\left(a+4\right)$. Multiply $\frac{19}{\left(a-4\right)\left(a+4\right)}$ times $\frac{4}{4}$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19\times 4}{4\left(a-4\right)\left(a+4\right)}$$

Since $\frac{-64}{4\left(a-4\right)\left(a+4\right)}$ and $\frac{19\times 4}{4\left(a-4\right)\left(a+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-64+19\times 4}{4\left(a-4\right)\left(a+4\right)}$$

Do the multiplications in $-64+19\times 4$.

$$\frac{-64+76}{4\left(a-4\right)\left(a+4\right)}$$

Do the calculations in $-64+76$.

$$\frac{12}{4\left(a-4\right)\left(a+4\right)}$$

Expand $4\left(a-4\right)\left(a+4\right)$.

$$\frac{12}{4a^{2}-64}$$

Expand

$\frac{3}{a^{2}-16}$

Short Solution Steps

Factor $4a+16$. Factor $4a-16$.

$$\frac{a+12}{4\left(a+4\right)}-\frac{a+4}{4\left(a-4\right)}+\frac{19}{a^{2}-16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(a+4\right)$ and $4\left(a-4\right)$ is $4\left(a-4\right)\left(a+4\right)$. Multiply $\frac{a+12}{4\left(a+4\right)}$ times $\frac{a-4}{a-4}$. Multiply $\frac{a+4}{4\left(a-4\right)}$ times $\frac{a+4}{a+4}$.

$$\frac{\left(a+12\right)\left(a-4\right)}{4\left(a-4\right)\left(a+4\right)}-\frac{\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Since $\frac{\left(a+12\right)\left(a-4\right)}{4\left(a-4\right)\left(a+4\right)}$ and $\frac{\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+12\right)\left(a-4\right)-\left(a+4\right)\left(a+4\right)}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Do the multiplications in $\left(a+12\right)\left(a-4\right)-\left(a+4\right)\left(a+4\right)$.

$$\frac{a^{2}-4a+12a-48-a^{2}-4a-4a-16}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Combine like terms in $a^{2}-4a+12a-48-a^{2}-4a-4a-16$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{a^{2}-16}$$

Factor $a^{2}-16$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19}{\left(a-4\right)\left(a+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(a-4\right)\left(a+4\right)$ and $\left(a-4\right)\left(a+4\right)$ is $4\left(a-4\right)\left(a+4\right)$. Multiply $\frac{19}{\left(a-4\right)\left(a+4\right)}$ times $\frac{4}{4}$.

$$\frac{-64}{4\left(a-4\right)\left(a+4\right)}+\frac{19\times 4}{4\left(a-4\right)\left(a+4\right)}$$

Since $\frac{-64}{4\left(a-4\right)\left(a+4\right)}$ and $\frac{19\times 4}{4\left(a-4\right)\left(a+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-64+19\times 4}{4\left(a-4\right)\left(a+4\right)}$$

Do the multiplications in $-64+19\times 4$.

$$\frac{-64+76}{4\left(a-4\right)\left(a+4\right)}$$

Do the calculations in $-64+76$.

$$\frac{12}{4\left(a-4\right)\left(a+4\right)}$$

Expand $4\left(a-4\right)\left(a+4\right)$.

$$\frac{12}{4a^{2}-64}$$