Solve x^2+2x^2+5x+6=0

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

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{47}i}{6}\approx -0.833333333+1.1426091i$$

$$x=\frac{-\sqrt{47}i-5}{6}\approx -0.833333333-1.1426091i$$

Steps Using the Quadratic Formula

Combine x^{2} and 2x^{2} to get 3x^{2}.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\times 6}}{2\times 3}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 3\times 6}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-5±\sqrt{25-12\times 6}}{2\times 3}$$

Multiply -12 times 6.

$$x=\frac{-5±\sqrt{25-72}}{2\times 3}$$

Add 25 to -72.

$$x=\frac{-5±\sqrt{-47}}{2\times 3}$$

Take the square root of -47.

$$x=\frac{-5±\sqrt{47}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-5±\sqrt{47}i}{6}$$

Now solve the equation x=\frac{-5±\sqrt{47}i}{6} when ± is plus. Add -5 to i\sqrt{47}.

$$x=\frac{-5+\sqrt{47}i}{6}$$

Now solve the equation x=\frac{-5±\sqrt{47}i}{6} when ± is minus. Subtract i\sqrt{47} from -5.

$$x=\frac{-\sqrt{47}i-5}{6}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-5}{6}$$

Steps for Completing the Square

Combine x^{2} and 2x^{2} to get 3x^{2}.

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

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by 3.

$$\frac{3x^{2}+5x}{3}=-\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{5}{3}x=-\frac{6}{3}$$

Divide -6 by 3.

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

Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=-2+\frac{25}{36}$$

Add -2 to \frac{25}{36}.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=-\frac{47}{36}$$

Factor x^{2}+\frac{5}{3}x+\frac{25}{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}.

$$\left(x+\frac{5}{6}\right)^{2}=-\frac{47}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}$$

Simplify.

$$x+\frac{5}{6}=\frac{\sqrt{47}i}{6} x+\frac{5}{6}=-\frac{\sqrt{47}i}{6}$$

Subtract \frac{5}{6} from both sides of the equation.

$$x=\frac{-5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-5}{6}$$