Solve (3)/(5x)-(x-1)/(3(x)^(2))-((x)^(2)+2x+3)/(15(x)^(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

$$\frac{ 3 }{ 5x } - \frac{ x-1 }{ 3 { x }^{ 2 } } - \frac{ { x }^{ 2 } +2x+3 }{ 15 { x }^{ 3 } }$$

Evaluate

$\frac{x^{2}+x-1}{5x^{3}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $5x$ and $3x^{2}$ is $15x^{2}$. Multiply $\frac{3}{5x}$ times $\frac{3x}{3x}$. Multiply $\frac{x-1}{3x^{2}}$ times $\frac{5}{5}$.

$$\frac{3\times 3x}{15x^{2}}-\frac{5\left(x-1\right)}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Since $\frac{3\times 3x}{15x^{2}}$ and $\frac{5\left(x-1\right)}{15x^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 3x-5\left(x-1\right)}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Do the multiplications in $3\times 3x-5\left(x-1\right)$.

$$\frac{9x-5x+5}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Combine like terms in $9x-5x+5$.

$$\frac{4x+5}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $15x^{2}$ and $15x^{3}$ is $15x^{3}$. Multiply $\frac{4x+5}{15x^{2}}$ times $\frac{x}{x}$.

$$\frac{\left(4x+5\right)x}{15x^{3}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Since $\frac{\left(4x+5\right)x}{15x^{3}}$ and $\frac{x^{2}+2x+3}{15x^{3}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(4x+5\right)x-\left(x^{2}+2x+3\right)}{15x^{3}}$$

Do the multiplications in $\left(4x+5\right)x-\left(x^{2}+2x+3\right)$.

$$\frac{4x^{2}+5x-x^{2}-2x-3}{15x^{3}}$$

Combine like terms in $4x^{2}+5x-x^{2}-2x-3$.

$$\frac{3x^{2}+3x-3}{15x^{3}}$$

Factor the expressions that are not already factored in $\frac{3x^{2}+3x-3}{15x^{3}}$.

$$\frac{3\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{15x^{3}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{5x^{3}}$$

To find the opposite of $-\frac{1}{2}\sqrt{5}-\frac{1}{2}$, find the opposite of each term.

$$\frac{\left(x+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{5x^{3}}$$

To find the opposite of $\frac{1}{2}\sqrt{5}-\frac{1}{2}$, find the opposite of each term.

$$\frac{\left(x+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)}{5x^{3}}$$

Use the distributive property to multiply $x+\frac{1}{2}\sqrt{5}+\frac{1}{2}$ by $x-\frac{1}{2}\sqrt{5}+\frac{1}{2}$ and combine like terms.

$$\frac{x^{2}+x-\frac{1}{4}\left(\sqrt{5}\right)^{2}+\frac{1}{4}}{5x^{3}}$$

The square of $\sqrt{5}$ is $5$.

$$\frac{x^{2}+x-\frac{1}{4}\times 5+\frac{1}{4}}{5x^{3}}$$

Multiply $-\frac{1}{4}$ and $5$ to get $-\frac{5}{4}$.

$$\frac{x^{2}+x-\frac{5}{4}+\frac{1}{4}}{5x^{3}}$$

Add $-\frac{5}{4}$ and $\frac{1}{4}$ to get $-1$.

$$\frac{x^{2}+x-1}{5x^{3}}$$

Expand

$\frac{x^{2}+x-1}{5x^{3}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $5x$ and $3x^{2}$ is $15x^{2}$. Multiply $\frac{3}{5x}$ times $\frac{3x}{3x}$. Multiply $\frac{x-1}{3x^{2}}$ times $\frac{5}{5}$.

$$\frac{3\times 3x}{15x^{2}}-\frac{5\left(x-1\right)}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Since $\frac{3\times 3x}{15x^{2}}$ and $\frac{5\left(x-1\right)}{15x^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 3x-5\left(x-1\right)}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Do the multiplications in $3\times 3x-5\left(x-1\right)$.

$$\frac{9x-5x+5}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Combine like terms in $9x-5x+5$.

$$\frac{4x+5}{15x^{2}}-\frac{x^{2}+2x+3}{15x^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $15x^{2}$ and $15x^{3}$ is $15x^{3}$. Multiply $\frac{4x+5}{15x^{2}}$ times $\frac{x}{x}$.

$$\frac{\left(4x+5\right)x}{15x^{3}}-\frac{x^{2}+2x+3}{15x^{3}}$$

Since $\frac{\left(4x+5\right)x}{15x^{3}}$ and $\frac{x^{2}+2x+3}{15x^{3}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(4x+5\right)x-\left(x^{2}+2x+3\right)}{15x^{3}}$$

Do the multiplications in $\left(4x+5\right)x-\left(x^{2}+2x+3\right)$.

$$\frac{4x^{2}+5x-x^{2}-2x-3}{15x^{3}}$$

Combine like terms in $4x^{2}+5x-x^{2}-2x-3$.

$$\frac{3x^{2}+3x-3}{15x^{3}}$$

Factor the expressions that are not already factored in $\frac{3x^{2}+3x-3}{15x^{3}}$.

$$\frac{3\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{15x^{3}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{5x^{3}}$$

To find the opposite of $-\frac{1}{2}\sqrt{5}-\frac{1}{2}$, find the opposite of each term.

$$\frac{\left(x+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{5x^{3}}$$

To find the opposite of $\frac{1}{2}\sqrt{5}-\frac{1}{2}$, find the opposite of each term.

$$\frac{\left(x+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)}{5x^{3}}$$

Use the distributive property to multiply $x+\frac{1}{2}\sqrt{5}+\frac{1}{2}$ by $x-\frac{1}{2}\sqrt{5}+\frac{1}{2}$ and combine like terms.

$$\frac{x^{2}+x-\frac{1}{4}\left(\sqrt{5}\right)^{2}+\frac{1}{4}}{5x^{3}}$$

The square of $\sqrt{5}$ is $5$.

$$\frac{x^{2}+x-\frac{1}{4}\times 5+\frac{1}{4}}{5x^{3}}$$

Multiply $-\frac{1}{4}$ and $5$ to get $-\frac{5}{4}$.

$$\frac{x^{2}+x-\frac{5}{4}+\frac{1}{4}}{5x^{3}}$$

Add $-\frac{5}{4}$ and $\frac{1}{4}$ to get $-1$.

$$\frac{x^{2}+x-1}{5x^{3}}$$