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

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

Solve for x

$$x = \frac{\sqrt{21} + 1}{2} \approx 2.791287847$$

$$x=\frac{1-\sqrt{21}}{2}\approx -1.791287847$$

Steps Using the Quadratic Formula

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

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

Subtract 3 from both sides.

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

Subtract 3 from -2 to get -5.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$x=\frac{-\left(-1\right)±\sqrt{1+20}}{2}$$

Add 1 to 20.

$$x=\frac{-\left(-1\right)±\sqrt{21}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{21}}{2}$$

Now solve the equation x=\frac{1±\sqrt{21}}{2} when ± is plus. Add 1 to \sqrt{21}.

$$x=\frac{\sqrt{21}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{21}}{2} when ± is minus. Subtract \sqrt{21} from 1.

$$x=\frac{1-\sqrt{21}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{21}+1}{2} x=\frac{1-\sqrt{21}}{2}$$

Steps for Completing the Square

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

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

Add 2 to both sides.

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

Add 3 and 2 to get 5.

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

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=5+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=5+\frac{1}{4}$$

Add 5 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{21}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{21}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{21}}{2} x-\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{21}+1}{2} x=\frac{1-\sqrt{21}}{2}$$