Solve (a - 1)/(3a ^ 2) + (a + 3)/(4a) | 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

$$\frac{a-1}{3a^{2}}+\frac{a+3}{4a}$$

Evaluate

$\frac{1}{4}+\frac{13}{12a}-\frac{1}{3a^{2}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3a^{2}$ and $4a$ is $12a^{2}$. Multiply $\frac{a-1}{3a^{2}}$ times $\frac{4}{4}$. Multiply $\frac{a+3}{4a}$ times $\frac{3a}{3a}$.

$$\frac{4\left(a-1\right)}{12a^{2}}+\frac{\left(a+3\right)\times 3a}{12a^{2}}$$

Since $\frac{4\left(a-1\right)}{12a^{2}}$ and $\frac{\left(a+3\right)\times 3a}{12a^{2}}$ have the same denominator, add them by adding their numerators.

$$\frac{4\left(a-1\right)+\left(a+3\right)\times 3a}{12a^{2}}$$

Do the multiplications in $4\left(a-1\right)+\left(a+3\right)\times 3a$.

$$\frac{4a-4+3a^{2}+9a}{12a^{2}}$$

Combine like terms in $4a-4+3a^{2}+9a$.

$$\frac{13a-4+3a^{2}}{12a^{2}}$$

Factor the expressions that are not already factored in $\frac{13a-4+3a^{2}}{12a^{2}}$.

$$\frac{3\left(a-\left(-\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{12a^{2}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{\left(a-\left(-\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{4a^{2}}$$

To find the opposite of $-\frac{1}{6}\sqrt{217}-\frac{13}{6}$, find the opposite of each term.

$$\frac{\left(a+\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{4a^{2}}$$

To find the opposite of $\frac{1}{6}\sqrt{217}-\frac{13}{6}$, find the opposite of each term.

$$\frac{\left(a+\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)\left(a-\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)}{4a^{2}}$$

Use the distributive property to multiply $a+\frac{1}{6}\sqrt{217}+\frac{13}{6}$ by $a-\frac{1}{6}\sqrt{217}+\frac{13}{6}$ and combine like terms.

$$\frac{a^{2}+\frac{13}{3}a-\frac{1}{36}\left(\sqrt{217}\right)^{2}+\frac{169}{36}}{4a^{2}}$$

The square of $\sqrt{217}$ is $217$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{1}{36}\times 217+\frac{169}{36}}{4a^{2}}$$

Multiply $-\frac{1}{36}$ and $217$ to get $-\frac{217}{36}$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{217}{36}+\frac{169}{36}}{4a^{2}}$$

Add $-\frac{217}{36}$ and $\frac{169}{36}$ to get $-\frac{4}{3}$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{4}{3}}{4a^{2}}$$

Expand

$\frac{1}{4}+\frac{13}{12a}-\frac{1}{3a^{2}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3a^{2}$ and $4a$ is $12a^{2}$. Multiply $\frac{a-1}{3a^{2}}$ times $\frac{4}{4}$. Multiply $\frac{a+3}{4a}$ times $\frac{3a}{3a}$.

$$\frac{4\left(a-1\right)}{12a^{2}}+\frac{\left(a+3\right)\times 3a}{12a^{2}}$$

Since $\frac{4\left(a-1\right)}{12a^{2}}$ and $\frac{\left(a+3\right)\times 3a}{12a^{2}}$ have the same denominator, add them by adding their numerators.

$$\frac{4\left(a-1\right)+\left(a+3\right)\times 3a}{12a^{2}}$$

Do the multiplications in $4\left(a-1\right)+\left(a+3\right)\times 3a$.

$$\frac{4a-4+3a^{2}+9a}{12a^{2}}$$

Combine like terms in $4a-4+3a^{2}+9a$.

$$\frac{13a-4+3a^{2}}{12a^{2}}$$

Factor the expressions that are not already factored in $\frac{13a-4+3a^{2}}{12a^{2}}$.

$$\frac{3\left(a-\left(-\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{12a^{2}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{\left(a-\left(-\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{4a^{2}}$$

To find the opposite of $-\frac{1}{6}\sqrt{217}-\frac{13}{6}$, find the opposite of each term.

$$\frac{\left(a+\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)\left(a-\left(\frac{1}{6}\sqrt{217}-\frac{13}{6}\right)\right)}{4a^{2}}$$

To find the opposite of $\frac{1}{6}\sqrt{217}-\frac{13}{6}$, find the opposite of each term.

$$\frac{\left(a+\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)\left(a-\frac{1}{6}\sqrt{217}+\frac{13}{6}\right)}{4a^{2}}$$

Use the distributive property to multiply $a+\frac{1}{6}\sqrt{217}+\frac{13}{6}$ by $a-\frac{1}{6}\sqrt{217}+\frac{13}{6}$ and combine like terms.

$$\frac{a^{2}+\frac{13}{3}a-\frac{1}{36}\left(\sqrt{217}\right)^{2}+\frac{169}{36}}{4a^{2}}$$

The square of $\sqrt{217}$ is $217$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{1}{36}\times 217+\frac{169}{36}}{4a^{2}}$$

Multiply $-\frac{1}{36}$ and $217$ to get $-\frac{217}{36}$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{217}{36}+\frac{169}{36}}{4a^{2}}$$

Add $-\frac{217}{36}$ and $\frac{169}{36}$ to get $-\frac{4}{3}$.

$$\frac{a^{2}+\frac{13}{3}a-\frac{4}{3}}{4a^{2}}$$