Solve (12/5sqrt{3}x)^2+2x+1=0

$$( \frac { 12 } { 5 } \sqrt { 3 } x ) ^ { 2 } + 2 x + 1 = 0$$

Solve for x (complex solution)

$$x=\frac{-25+5\sqrt{407}i}{432}\approx -0.05787037+0.23349816i$$

$$x=\frac{-5\sqrt{407}i-25}{432}\approx -0.05787037-0.23349816i$$

Steps Using the Quadratic Formula

Expand \left(\frac{12}{5}\sqrt{3}x\right)^{2}.

$$\left(\frac{12}{5}\right)^{2}\left(\sqrt{3}\right)^{2}x^{2}+2x+1=0$$

Calculate \frac{12}{5} to the power of 2 and get \frac{144}{25}.

$$\frac{144}{25}\left(\sqrt{3}\right)^{2}x^{2}+2x+1=0$$

The square of \sqrt{3} is 3.

$$\frac{144}{25}\times 3x^{2}+2x+1=0$$

Multiply \frac{144}{25} and 3 to get \frac{432}{25}.

$$\frac{432}{25}x^{2}+2x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{432}{25} for a, 2 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times \frac{432}{25}}}{2\times \frac{432}{25}}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times \frac{432}{25}}}{2\times \frac{432}{25}}$$

Multiply -4 times \frac{432}{25}.

$$x=\frac{-2±\sqrt{4-\frac{1728}{25}}}{2\times \frac{432}{25}}$$

Add 4 to -\frac{1728}{25}.

$$x=\frac{-2±\sqrt{-\frac{1628}{25}}}{2\times \frac{432}{25}}$$

Take the square root of -\frac{1628}{25}.

$$x=\frac{-2±\frac{2\sqrt{407}i}{5}}{2\times \frac{432}{25}}$$

Multiply 2 times \frac{432}{25}.

$$x=\frac{-2±\frac{2\sqrt{407}i}{5}}{\frac{864}{25}}$$

Now solve the equation x=\frac{-2±\frac{2\sqrt{407}i}{5}}{\frac{864}{25}} when ± is plus. Add -2 to \frac{2i\sqrt{407}}{5}.

$$x=\frac{\frac{2\sqrt{407}i}{5}-2}{\frac{864}{25}}$$

Divide -2+\frac{2i\sqrt{407}}{5} by \frac{864}{25} by multiplying -2+\frac{2i\sqrt{407}}{5} by the reciprocal of \frac{864}{25}.

$$x=\frac{-25+5\sqrt{407}i}{432}$$

Now solve the equation x=\frac{-2±\frac{2\sqrt{407}i}{5}}{\frac{864}{25}} when ± is minus. Subtract \frac{2i\sqrt{407}}{5} from -2.

$$x=\frac{-\frac{2\sqrt{407}i}{5}-2}{\frac{864}{25}}$$

Divide -2-\frac{2i\sqrt{407}}{5} by \frac{864}{25} by multiplying -2-\frac{2i\sqrt{407}}{5} by the reciprocal of \frac{864}{25}.

$$x=\frac{-5\sqrt{407}i-25}{432}$$

The equation is now solved.

$$x=\frac{-25+5\sqrt{407}i}{432} x=\frac{-5\sqrt{407}i-25}{432}$$

Steps for Completing the Square

Expand \left(\frac{12}{5}\sqrt{3}x\right)^{2}.

$$\left(\frac{12}{5}\right)^{2}\left(\sqrt{3}\right)^{2}x^{2}+2x+1=0$$

Calculate \frac{12}{5} to the power of 2 and get \frac{144}{25}.

$$\frac{144}{25}\left(\sqrt{3}\right)^{2}x^{2}+2x+1=0$$

The square of \sqrt{3} is 3.

$$\frac{144}{25}\times 3x^{2}+2x+1=0$$

Multiply \frac{144}{25} and 3 to get \frac{432}{25}.

$$\frac{432}{25}x^{2}+2x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$\frac{432}{25}x^{2}+2x=-1$$

Divide both sides of the equation by \frac{432}{25}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{432}{25}x^{2}+2x}{\frac{432}{25}}=-\frac{1}{\frac{432}{25}}$$

Dividing by \frac{432}{25} undoes the multiplication by \frac{432}{25}.

$$x^{2}+\frac{2}{\frac{432}{25}}x=-\frac{1}{\frac{432}{25}}$$

Divide 2 by \frac{432}{25} by multiplying 2 by the reciprocal of \frac{432}{25}.

$$x^{2}+\frac{25}{216}x=-\frac{1}{\frac{432}{25}}$$

Divide -1 by \frac{432}{25} by multiplying -1 by the reciprocal of \frac{432}{25}.

$$x^{2}+\frac{25}{216}x=-\frac{25}{432}$$

Divide \frac{25}{216}, the coefficient of the x term, by 2 to get \frac{25}{432}. Then add the square of \frac{25}{432} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{216}x+\left(\frac{25}{432}\right)^{2}=-\frac{25}{432}+\left(\frac{25}{432}\right)^{2}$$

Square \frac{25}{432} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{216}x+\frac{625}{186624}=-\frac{25}{432}+\frac{625}{186624}$$

Add -\frac{25}{432} to \frac{625}{186624} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{216}x+\frac{625}{186624}=-\frac{10175}{186624}$$

Factor x^{2}+\frac{25}{216}x+\frac{625}{186624}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{432}\right)^{2}=-\frac{10175}{186624}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{432}\right)^{2}}=\sqrt{-\frac{10175}{186624}}$$

Simplify.

$$x+\frac{25}{432}=\frac{5\sqrt{407}i}{432} x+\frac{25}{432}=-\frac{5\sqrt{407}i}{432}$$

Subtract \frac{25}{432} from both sides of the equation.

$$x=\frac{-25+5\sqrt{407}i}{432} x=\frac{-5\sqrt{407}i-25}{432}$$