Solve (x-1)(x+3)+5=0

$$( x - 1 ) ( x + 3 ) + 5 = 0$$

Solve for x (complex solution)

$$x=-1+i$$

$$x=-1-i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$x^{2}+2x-3+5=0$$

Add -3 and 5 to get 2.

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

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

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

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8}}{2}$$

Add 4 to -8.

$$x=\frac{-2±\sqrt{-4}}{2}$$

Take the square root of -4.

$$x=\frac{-2±2i}{2}$$

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

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

Divide -2+2i by 2.

$$x=-1+i$$

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

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

Divide -2-2i by 2.

$$x=-1-i$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$x^{2}+2x-3+5=0$$

Add -3 and 5 to get 2.

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Square 1.

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

Add -2 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 1 from both sides of the equation.

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