Solve 2x+5=-x^2

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

Solve for x (complex solution)

$$x=-1+2i$$

$$x=-1-2i$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

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

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.

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

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

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

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 5}}{2}$$

Multiply -4 times 5.

$$x=\frac{-2±\sqrt{4-20}}{2}$$

Add 4 to -20.

$$x=\frac{-2±\sqrt{-16}}{2}$$

Take the square root of -16.

$$x=\frac{-2±4i}{2}$$

Now solve the equation x=\frac{-2±4i}{2} when ± is plus. Add -2 to 4i.

$$x=\frac{-2+4i}{2}$$

Divide -2+4i by 2.

$$x=-1+2i$$

Now solve the equation x=\frac{-2±4i}{2} when ± is minus. Subtract 4i from -2.

$$x=\frac{-2-4i}{2}$$

Divide -2-4i by 2.

$$x=-1-2i$$

The equation is now solved.

$$x=-1+2i x=-1-2i$$

Steps for Completing the Square

Add x^{2} to both sides.

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

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

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

$$x^{2}+2x+1^{2}=-5+1^{2}$$

Square 1.

$$x^{2}+2x+1=-5+1$$

Add -5 to 1.

$$x^{2}+2x+1=-4$$

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

$$\left(x+1\right)^{2}=-4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-4}$$

Simplify.

$$x+1=2i x+1=-2i$$

Subtract 1 from both sides of the equation.

$$x=-1+2i x=-1-2i$$