Solve -375=(x+1)^2-4

$$- 375 = ( x + 1 ) ^ { 2 } - 4$$

Solve for x (complex solution)

$$x=-\sqrt{371}i-1\approx -1-19.261360284i$$

$$x=-1+\sqrt{371}i\approx -1+19.261360284i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$-375=x^{2}+2x+1-4$$

Subtract 4 from 1 to get -3.

$$-375=x^{2}+2x-3$$

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

$$x^{2}+2x-3=-375$$

Add 375 to both sides.

$$x^{2}+2x-3+375=0$$

Add -3 and 375 to get 372.

$$x^{2}+2x+372=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 372 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 372}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 372}}{2}$$

Multiply -4 times 372.

$$x=\frac{-2±\sqrt{4-1488}}{2}$$

Add 4 to -1488.

$$x=\frac{-2±\sqrt{-1484}}{2}$$

Take the square root of -1484.

$$x=\frac{-2±2\sqrt{371}i}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{371}i}{2} when ± is plus. Add -2 to 2i\sqrt{371}.

$$x=\frac{-2+2\sqrt{371}i}{2}$$

Divide -2+2i\sqrt{371} by 2.

$$x=-1+\sqrt{371}i$$

Now solve the equation x=\frac{-2±2\sqrt{371}i}{2} when ± is minus. Subtract 2i\sqrt{371} from -2.

$$x=\frac{-2\sqrt{371}i-2}{2}$$

Divide -2-2i\sqrt{371} by 2.

$$x=-\sqrt{371}i-1$$

The equation is now solved.

$$x=-1+\sqrt{371}i x=-\sqrt{371}i-1$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$-375=x^{2}+2x+1-4$$

Subtract 4 from 1 to get -3.

$$-375=x^{2}+2x-3$$

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

$$x^{2}+2x-3=-375$$

Add 3 to both sides.

$$x^{2}+2x=-375+3$$

Add -375 and 3 to get -372.

$$x^{2}+2x=-372$$

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.

$$x^{2}+2x+1^{2}=-372+1^{2}$$

Square 1.

$$x^{2}+2x+1=-372+1$$

Add -372 to 1.

$$x^{2}+2x+1=-371$$

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}.

$$\left(x+1\right)^{2}=-371$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-371}$$

Simplify.

$$x+1=\sqrt{371}i x+1=-\sqrt{371}i$$

Subtract 1 from both sides of the equation.

$$x=-1+\sqrt{371}i x=-\sqrt{371}i-1$$