Solve left(3xright)^2+4x+2=0

$${ \left(3x \right) }^{ 2 } +4x+2=0$$

Solve for x (complex solution)

$$x=\frac{-2+\sqrt{14}i}{9}\approx -0.222222222+0.41573971i$$

$$x=\frac{-\sqrt{14}i-2}{9}\approx -0.222222222-0.41573971i$$

Steps Using the Quadratic Formula

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

$$3^{2}x^{2}+4x+2=0$$

Calculate 3 to the power of 2 and get 9.

$$9x^{2}+4x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 4 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 9\times 2}}{2\times 9}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 9\times 2}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-4±\sqrt{16-36\times 2}}{2\times 9}$$

Multiply -36 times 2.

$$x=\frac{-4±\sqrt{16-72}}{2\times 9}$$

Add 16 to -72.

$$x=\frac{-4±\sqrt{-56}}{2\times 9}$$

Take the square root of -56.

$$x=\frac{-4±2\sqrt{14}i}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-4±2\sqrt{14}i}{18}$$

Now solve the equation x=\frac{-4±2\sqrt{14}i}{18} when ± is plus. Add -4 to 2i\sqrt{14}.

$$x=\frac{-4+2\sqrt{14}i}{18}$$

Divide -4+2i\sqrt{14} by 18.

$$x=\frac{-2+\sqrt{14}i}{9}$$

Now solve the equation x=\frac{-4±2\sqrt{14}i}{18} when ± is minus. Subtract 2i\sqrt{14} from -4.

$$x=\frac{-2\sqrt{14}i-4}{18}$$

Divide -4-2i\sqrt{14} by 18.

$$x=\frac{-\sqrt{14}i-2}{9}$$

The equation is now solved.

$$x=\frac{-2+\sqrt{14}i}{9} x=\frac{-\sqrt{14}i-2}{9}$$

Steps for Completing the Square

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

$$3^{2}x^{2}+4x+2=0$$

Calculate 3 to the power of 2 and get 9.

$$9x^{2}+4x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}+4x=-2$$

Divide both sides by 9.

$$\frac{9x^{2}+4x}{9}=-\frac{2}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{4}{9}x=-\frac{2}{9}$$

Divide \frac{4}{9}, the coefficient of the x term, by 2 to get \frac{2}{9}. Then add the square of \frac{2}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{9}x+\left(\frac{2}{9}\right)^{2}=-\frac{2}{9}+\left(\frac{2}{9}\right)^{2}$$

Square \frac{2}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=-\frac{2}{9}+\frac{4}{81}$$

Add -\frac{2}{9} to \frac{4}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=-\frac{14}{81}$$

Factor x^{2}+\frac{4}{9}x+\frac{4}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{9}\right)^{2}=-\frac{14}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{9}\right)^{2}}=\sqrt{-\frac{14}{81}}$$

Simplify.

$$x+\frac{2}{9}=\frac{\sqrt{14}i}{9} x+\frac{2}{9}=-\frac{\sqrt{14}i}{9}$$

Subtract \frac{2}{9} from both sides of the equation.

$$x=\frac{-2+\sqrt{14}i}{9} x=\frac{-\sqrt{14}i-2}{9}$$