Solve 6x+61+18x^2=0

$$6 x + 61 + 18 x ^ { 2 } = 0$$

Solve for x (complex solution)

$$x=-\frac{1}{6}+\frac{11}{6}i\approx -0.166666667+1.833333333i$$

$$x=-\frac{1}{6}-\frac{11}{6}i\approx -0.166666667-1.833333333i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$18x^{2}+6x+61=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 6 for b, and 61 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 18\times 61}}{2\times 18}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 18\times 61}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-6±\sqrt{36-72\times 61}}{2\times 18}$$

Multiply -72 times 61.

$$x=\frac{-6±\sqrt{36-4392}}{2\times 18}$$

Add 36 to -4392.

$$x=\frac{-6±\sqrt{-4356}}{2\times 18}$$

Take the square root of -4356.

$$x=\frac{-6±66i}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-6±66i}{36}$$

Now solve the equation x=\frac{-6±66i}{36} when ± is plus. Add -6 to 66i.

$$x=\frac{-6+66i}{36}$$

Divide -6+66i by 36.

$$x=-\frac{1}{6}+\frac{11}{6}i$$

Now solve the equation x=\frac{-6±66i}{36} when ± is minus. Subtract 66i from -6.

$$x=\frac{-6-66i}{36}$$

Divide -6-66i by 36.

$$x=-\frac{1}{6}-\frac{11}{6}i$$

The equation is now solved.

$$x=-\frac{1}{6}+\frac{11}{6}i x=-\frac{1}{6}-\frac{11}{6}i$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$18x^{2}+6x+61=0$$

Subtract 61 from both sides of the equation.

$$18x^{2}+6x+61-61=-61$$

Subtracting 61 from itself leaves 0.

$$18x^{2}+6x=-61$$

Divide both sides by 18.

$$\frac{18x^{2}+6x}{18}=-\frac{61}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{6}{18}x=-\frac{61}{18}$$

Reduce the fraction \frac{6}{18} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{1}{3}x=-\frac{61}{18}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{61}{18}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{61}{18}+\frac{1}{36}$$

Add -\frac{61}{18} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{121}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{121}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{11}{6}i x+\frac{1}{6}=-\frac{11}{6}i$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=-\frac{1}{6}+\frac{11}{6}i x=-\frac{1}{6}-\frac{11}{6}i$$