Solve x^2+3x+2x+7=0

$$x ^ { 2 } + 3 x + 2 x + 7 = 0$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{3}i}{2}\approx -2.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i-5}{2}\approx -2.5-0.866025404i$$

Steps Using the Quadratic Formula

Combine 3x and 2x to get 5x.

$$x^{2}+5x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 7}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-5±\sqrt{25-28}}{2}$$

Add 25 to -28.

$$x=\frac{-5±\sqrt{-3}}{2}$$

Take the square root of -3.

$$x=\frac{-5±\sqrt{3}i}{2}$$

Now solve the equation x=\frac{-5±\sqrt{3}i}{2} when ± is plus. Add -5 to i\sqrt{3}.

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

Now solve the equation x=\frac{-5±\sqrt{3}i}{2} when ± is minus. Subtract i\sqrt{3} from -5.

$$x=\frac{-\sqrt{3}i-5}{2}$$

The equation is now solved.

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

Steps for Completing the Square

Combine 3x and 2x to get 5x.

$$x^{2}+5x+7=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+5x=-7$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-7+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-7+\frac{25}{4}$$

Add -7 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=-\frac{3}{4}$$

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

$$\left(x+\frac{5}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

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

Subtract \frac{5}{2} from both sides of the equation.

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