1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve + ( 2 | Equatix

Question

$$\frac { 2 - ( \frac { 2 } { \sqrt { 3 } } ) ^ { 2 } } { 2 + ( \frac { 2 } { \sqrt { 3 } } ) ^ { 2 } }$$

Evaluate

$\frac{1}{5}=0.2$

Solution Steps

Rationalize the denominator of $\frac{2}{\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.

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

The square of $\sqrt{3}$ is $3$.

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

To raise $\frac{2\sqrt{3}}{3}$ to a power, raise both numerator and denominator to the power and then divide.

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

Expand $\left(2\sqrt{3}\right)^{2}$.

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

Calculate $2$ to the power of $2$ and get $4$.

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

The square of $\sqrt{3}$ is $3$.

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

Multiply $4$ and $3$ to get $12$.

$$\frac{2-\frac{12}{3^{2}}}{2+\left(\frac{2}{\sqrt{3}}\right)^{2}}$$

Calculate $3$ to the power of $2$ and get $9$.

$$\frac{2-\frac{12}{9}}{2+\left(\frac{2}{\sqrt{3}}\right)^{2}}$$

Reduce the fraction $\frac{12}{9}$ to lowest terms by extracting and canceling out $3$.

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

Subtract $\frac{4}{3}$ from $2$ to get $\frac{2}{3}$.

$$\frac{\frac{2}{3}}{2+\left(\frac{2}{\sqrt{3}}\right)^{2}}$$

Rationalize the denominator of $\frac{2}{\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.

$$\frac{\frac{2}{3}}{2+\left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{2}}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{\frac{2}{3}}{2+\left(\frac{2\sqrt{3}}{3}\right)^{2}}$$

To raise $\frac{2\sqrt{3}}{3}$ to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{2}{3}}{2+\frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $2$ times $\frac{3^{2}}{3^{2}}$.

$$\frac{\frac{2}{3}}{\frac{2\times 3^{2}}{3^{2}}+\frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}}$$

Since $\frac{2\times 3^{2}}{3^{2}}$ and $\frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{2}{3}}{\frac{2\times 3^{2}+\left(2\sqrt{3}\right)^{2}}{3^{2}}}$$

Divide $\frac{2}{3}$ by $\frac{2\times 3^{2}+\left(2\sqrt{3}\right)^{2}}{3^{2}}$ by multiplying $\frac{2}{3}$ by the reciprocal of $\frac{2\times 3^{2}+\left(2\sqrt{3}\right)^{2}}{3^{2}}$.

$$\frac{2\times 3^{2}}{3\left(2\times 3^{2}+\left(2\sqrt{3}\right)^{2}\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{2\times 3}{\left(2\sqrt{3}\right)^{2}+2\times 3^{2}}$$

Multiply $2$ and $3$ to get $6$.

$$\frac{6}{\left(2\sqrt{3}\right)^{2}+2\times 3^{2}}$$

Expand $\left(2\sqrt{3}\right)^{2}$.

$$\frac{6}{2^{2}\left(\sqrt{3}\right)^{2}+2\times 3^{2}}$$

Calculate $2$ to the power of $2$ and get $4$.

$$\frac{6}{4\left(\sqrt{3}\right)^{2}+2\times 3^{2}}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{6}{4\times 3+2\times 3^{2}}$$

Multiply $4$ and $3$ to get $12$.

$$\frac{6}{12+2\times 3^{2}}$$

Calculate $3$ to the power of $2$ and get $9$.

$$\frac{6}{12+2\times 9}$$

Multiply $2$ and $9$ to get $18$.

$$\frac{6}{12+18}$$

Add $12$ and $18$ to get $30$.

$$\frac{6}{30}$$

Reduce the fraction $\frac{6}{30}$ to lowest terms by extracting and canceling out $6$.

$$\frac{1}{5}$$

Show Solution

Factor

$\frac{1}{5} = 0.2$

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