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

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

Solve for x

$$x = \frac{\sqrt{33} + 1}{4} \approx 1.686140662$$

$$x=\frac{1-\sqrt{33}}{4}\approx -1.186140662$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by 2x-3 and combine like terms.

$$2x^{2}-x-3=1$$

Subtract 1 from both sides.

$$2x^{2}-x-3-1=0$$

Subtract 1 from -3 to get -4.

$$2x^{2}-x-4=0$$

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

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

Multiply -4 times 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-\left(-1\right)±\sqrt{1+32}}{2\times 2}$$

Add 1 to 32.

$$x=\frac{-\left(-1\right)±\sqrt{33}}{2\times 2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{33}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1±\sqrt{33}}{4}$$

Now solve the equation x=\frac{1±\sqrt{33}}{4} when ± is plus. Add 1 to \sqrt{33}.

$$x=\frac{\sqrt{33}+1}{4}$$

Now solve the equation x=\frac{1±\sqrt{33}}{4} when ± is minus. Subtract \sqrt{33} from 1.

$$x=\frac{1-\sqrt{33}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{33}+1}{4} x=\frac{1-\sqrt{33}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by 2x-3 and combine like terms.

$$2x^{2}-x-3=1$$

Add 3 to both sides.

$$2x^{2}-x=1+3$$

Add 1 and 3 to get 4.

$$2x^{2}-x=4$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 4 by 2.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=2+\frac{1}{16}$$

Add 2 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{33}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{33}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{33}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{33}}{4} x-\frac{1}{4}=-\frac{\sqrt{33}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{33}+1}{4} x=\frac{1-\sqrt{33}}{4}$$