Solve 0=x^2+3x+14

$$0 = x ^ { 2 } + 3 x + 14$$

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{47}i}{2}\approx -1.5+3.4278273i$$

$$x=\frac{-\sqrt{47}i-3}{2}\approx -1.5-3.4278273i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+3x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 14}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 14}}{2}$$

Multiply -4 times 14.

$$x=\frac{-3±\sqrt{9-56}}{2}$$

Add 9 to -56.

$$x=\frac{-3±\sqrt{-47}}{2}$$

Take the square root of -47.

$$x=\frac{-3±\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-3±\sqrt{47}i}{2} when ± is plus. Add -3 to i\sqrt{47}.

$$x=\frac{-3+\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-3±\sqrt{47}i}{2} when ± is minus. Subtract i\sqrt{47} from -3.

$$x=\frac{-\sqrt{47}i-3}{2}$$

The equation is now solved.

$$x=\frac{-3+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i-3}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+3x+14=0$$

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+3x=-14$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-14+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-14+\frac{9}{4}$$

Add -14 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=-\frac{47}{4}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{47}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{47}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{47}i}{2} x+\frac{3}{2}=-\frac{\sqrt{47}i}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{-3+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i-3}{2}$$