Solve left(6xright)^2+5x+3=0

$${ \left(6x \right) }^{ 2 } +5x+3 = 0$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{407}i}{72}\approx -0.069444444+0.280197792i$$

$$x=\frac{-\sqrt{407}i-5}{72}\approx -0.069444444-0.280197792i$$

Steps Using the Quadratic Formula

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+5x+3=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+5x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 5 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 36\times 3}}{2\times 36}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 36\times 3}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-5±\sqrt{25-144\times 3}}{2\times 36}$$

Multiply -144 times 3.

$$x=\frac{-5±\sqrt{25-432}}{2\times 36}$$

Add 25 to -432.

$$x=\frac{-5±\sqrt{-407}}{2\times 36}$$

Take the square root of -407.

$$x=\frac{-5±\sqrt{407}i}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-5±\sqrt{407}i}{72}$$

Now solve the equation x=\frac{-5±\sqrt{407}i}{72} when ± is plus. Add -5 to i\sqrt{407}.

$$x=\frac{-5+\sqrt{407}i}{72}$$

Now solve the equation x=\frac{-5±\sqrt{407}i}{72} when ± is minus. Subtract i\sqrt{407} from -5.

$$x=\frac{-\sqrt{407}i-5}{72}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{407}i}{72} x=\frac{-\sqrt{407}i-5}{72}$$

Steps for Completing the Square

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+5x+3=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+5x+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$36x^{2}+5x=-3$$

Divide both sides by 36.

$$\frac{36x^{2}+5x}{36}=-\frac{3}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\frac{5}{36}x=-\frac{3}{36}$$

Reduce the fraction \frac{-3}{36} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{5}{36}x=-\frac{1}{12}$$

Divide \frac{5}{36}, the coefficient of the x term, by 2 to get \frac{5}{72}. Then add the square of \frac{5}{72} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{36}x+\left(\frac{5}{72}\right)^{2}=-\frac{1}{12}+\left(\frac{5}{72}\right)^{2}$$

Square \frac{5}{72} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{36}x+\frac{25}{5184}=-\frac{1}{12}+\frac{25}{5184}$$

Add -\frac{1}{12} to \frac{25}{5184} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{36}x+\frac{25}{5184}=-\frac{407}{5184}$$

Factor x^{2}+\frac{5}{36}x+\frac{25}{5184}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{72}\right)^{2}=-\frac{407}{5184}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{72}\right)^{2}}=\sqrt{-\frac{407}{5184}}$$

Simplify.

$$x+\frac{5}{72}=\frac{\sqrt{407}i}{72} x+\frac{5}{72}=-\frac{\sqrt{407}i}{72}$$

Subtract \frac{5}{72} from both sides of the equation.

$$x=\frac{-5+\sqrt{407}i}{72} x=\frac{-\sqrt{407}i-5}{72}$$