Solve x^2+2x+5=-100

$$x ^ { 2 } + 2 x + 5 = - 100$$

Solve for x (complex solution)

$$x=-1+2\sqrt{26}i\approx -1+10.198039027i$$

$$x=-2\sqrt{26}i-1\approx -1-10.198039027i$$

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.

$$x^{2}+2x+5=-100$$

Add 100 to both sides of the equation.

$$x^{2}+2x+5-\left(-100\right)=-100-\left(-100\right)$$

Subtracting -100 from itself leaves 0.

$$x^{2}+2x+5-\left(-100\right)=0$$

Subtract -100 from 5.

$$x^{2}+2x+105=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 105 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 105}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 105}}{2}$$

Multiply -4 times 105.

$$x=\frac{-2±\sqrt{4-420}}{2}$$

Add 4 to -420.

$$x=\frac{-2±\sqrt{-416}}{2}$$

Take the square root of -416.

$$x=\frac{-2±4\sqrt{26}i}{2}$$

Now solve the equation x=\frac{-2±4\sqrt{26}i}{2} when ± is plus. Add -2 to 4i\sqrt{26}.

$$x=\frac{-2+4\sqrt{26}i}{2}$$

Divide -2+4i\sqrt{26} by 2.

$$x=-1+2\sqrt{26}i$$

Now solve the equation x=\frac{-2±4\sqrt{26}i}{2} when ± is minus. Subtract 4i\sqrt{26} from -2.

$$x=\frac{-4\sqrt{26}i-2}{2}$$

Divide -2-4i\sqrt{26} by 2.

$$x=-2\sqrt{26}i-1$$

The equation is now solved.

$$x=-1+2\sqrt{26}i x=-2\sqrt{26}i-1$$

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.

$$x^{2}+2x+5=-100$$

Subtract 5 from both sides of the equation.

$$x^{2}+2x+5-5=-100-5$$

Subtracting 5 from itself leaves 0.

$$x^{2}+2x=-100-5$$

Subtract 5 from -100.

$$x^{2}+2x=-105$$

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.

$$x^{2}+2x+1^{2}=-105+1^{2}$$

Square 1.

$$x^{2}+2x+1=-105+1$$

Add -105 to 1.

$$x^{2}+2x+1=-104$$

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}.

$$\left(x+1\right)^{2}=-104$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-104}$$

Simplify.

$$x+1=2\sqrt{26}i x+1=-2\sqrt{26}i$$

Subtract 1 from both sides of the equation.

$$x=-1+2\sqrt{26}i x=-2\sqrt{26}i-1$$