Solve -((x-1)(x+2))=3

$$-((x-1)(x+2))=3$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$-\left(x^{2}+x-2\right)=3$$

To find the opposite of x^{2}+x-2, find the opposite of each term.

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

Subtract 3 from both sides.

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

Subtract 3 from 2 to get -1.

$$-x^{2}-x-1=0$$

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

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

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

Multiply 4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1-4}}{2\left(-1\right)}$$

Add 1 to -4.

$$x=\frac{-\left(-1\right)±\sqrt{-3}}{2\left(-1\right)}$$

Take the square root of -3.

$$x=\frac{-\left(-1\right)±\sqrt{3}i}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{3}i}{-2}$$

Now solve the equation x=\frac{1±\sqrt{3}i}{-2} when ± is plus. Add 1 to i\sqrt{3}.

$$x=\frac{1+\sqrt{3}i}{-2}$$

Divide 1+i\sqrt{3} by -2.

$$x=\frac{-\sqrt{3}i-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{3}i}{-2} when ± is minus. Subtract i\sqrt{3} from 1.

$$x=\frac{-\sqrt{3}i+1}{-2}$$

Divide 1-i\sqrt{3} by -2.

$$x=\frac{-1+\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{3}i-1}{2} x=\frac{-1+\sqrt{3}i}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$-\left(x^{2}+x-2\right)=3$$

To find the opposite of x^{2}+x-2, find the opposite of each term.

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

Subtract 2 from both sides.

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

Subtract 2 from 3 to get 1.

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

Divide both sides by -1.

$$\frac{-x^{2}-x}{-1}=\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

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

Divide -1 by -1.

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

Divide 1 by -1.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add -1 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{3}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{3}i}{2} x+\frac{1}{2}=-\frac{\sqrt{3}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i-1}{2}$$