Solve f(x)=1/3(sqrt{3}x-2)(sqrt{3}x+2)+2x

$$f ( x ) = \frac { 1 } { 3 } ( \sqrt { 3 } x - 2 ) ( \sqrt { 3 } x + 2 ) + 2 x$$

Evaluate

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

Solution Steps

Use the distributive property to multiply \frac{1}{3} by \sqrt{3}x-2.

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

Multiply \frac{1}{3} and -2 to get \frac{-2}{3}.

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

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

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

Apply the distributive property by multiplying each term of \frac{1}{3}\sqrt{3}x-\frac{2}{3} by each term of \sqrt{3}x+2.

$$\frac{1}{3}\sqrt{3}x\sqrt{3}x+\frac{1}{3}\sqrt{3}x\times 2-\frac{2}{3}\sqrt{3}x-\frac{2}{3}\times 2+2x$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{1}{3}\times 3xx+\frac{1}{3}\sqrt{3}x\times 2-\frac{2}{3}\sqrt{3}x-\frac{2}{3}\times 2+2x$$

Multiply x and x to get x^{2}.

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

Cancel out 3 and 3.

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

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$x^{2}+\frac{2}{3}\sqrt{3}x-\frac{2}{3}\sqrt{3}x-\frac{2}{3}\times 2+2x$$

Combine \frac{2}{3}\sqrt{3}x and -\frac{2}{3}\sqrt{3}x to get 0.

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

Express -\frac{2}{3}\times 2 as a single fraction.

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

Multiply -2 and 2 to get -4.

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

Fraction \frac{-4}{3} can be rewritten as -\frac{4}{3} by extracting the negative sign.

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

Factor

$$\left(x-\left(-\frac{\sqrt{21}}{3}-1\right)\right)\left(x-\left(\frac{\sqrt{21}}{3}-1\right)\right)$$