Solve x^2+2x+358=29

$${ x }^{ 2 } +2x+358=29$$

Solve for x (complex solution)

$$x=-1+2\sqrt{82}i\approx -1+18.110770276i$$

$$x=-2\sqrt{82}i-1\approx -1-18.110770276i$$

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.

$$x^{2}+2x+358=29$$

Subtract 29 from both sides of the equation.

$$x^{2}+2x+358-29=29-29$$

Subtracting 29 from itself leaves 0.

$$x^{2}+2x+358-29=0$$

Subtract 29 from 358.

$$x^{2}+2x+329=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 329 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 329}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 329}}{2}$$

Multiply -4 times 329.

$$x=\frac{-2±\sqrt{4-1316}}{2}$$

Add 4 to -1316.

$$x=\frac{-2±\sqrt{-1312}}{2}$$

Take the square root of -1312.

$$x=\frac{-2±4\sqrt{82}i}{2}$$

Now solve the equation x=\frac{-2±4\sqrt{82}i}{2} when ± is plus. Add -2 to 4i\sqrt{82}.

$$x=\frac{-2+4\sqrt{82}i}{2}$$

Divide -2+4i\sqrt{82} by 2.

$$x=-1+2\sqrt{82}i$$

Now solve the equation x=\frac{-2±4\sqrt{82}i}{2} when ± is minus. Subtract 4i\sqrt{82} from -2.

$$x=\frac{-4\sqrt{82}i-2}{2}$$

Divide -2-4i\sqrt{82} by 2.

$$x=-2\sqrt{82}i-1$$

The equation is now solved.

$$x=-1+2\sqrt{82}i x=-2\sqrt{82}i-1$$

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.

$$x^{2}+2x+358=29$$

Subtract 358 from both sides of the equation.

$$x^{2}+2x+358-358=29-358$$

Subtracting 358 from itself leaves 0.

$$x^{2}+2x=29-358$$

Subtract 358 from 29.

$$x^{2}+2x=-329$$

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.

$$x^{2}+2x+1^{2}=-329+1^{2}$$

Square 1.

$$x^{2}+2x+1=-329+1$$

Add -329 to 1.

$$x^{2}+2x+1=-328$$

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}.

$$\left(x+1\right)^{2}=-328$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-328}$$

Simplify.

$$x+1=2\sqrt{82}i x+1=-2\sqrt{82}i$$

Subtract 1 from both sides of the equation.

$$x=-1+2\sqrt{82}i x=-2\sqrt{82}i-1$$