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

$$\left( x+1 \right) \left( x-3 \right) - \left( x-2 \right) \left( 2x+1 \right) =1$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{7}i+1}{2}\approx 0.5-1.322875656i$$

$$x=\frac{1+\sqrt{7}i}{2}\approx 0.5+1.322875656i$$

Steps Using the Quadratic Formula

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

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

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

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

To find the opposite of 2x^{2}-3x-2, find the opposite of each term.

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

Combine x^{2} and -2x^{2} to get -x^{2}.

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

Combine -2x and 3x to get x.

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

Add -3 and 2 to get -1.

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

Subtract 1 from both sides.

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

Subtract 1 from -1 to get -2.

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

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

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times -2.

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

Add 1 to -8.

$$x=\frac{-1±\sqrt{-7}}{2\left(-1\right)}$$

Take the square root of -7.

$$x=\frac{-1±\sqrt{7}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\sqrt{7}i}{-2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{-2} when ± is plus. Add -1 to i\sqrt{7}.

$$x=\frac{-1+\sqrt{7}i}{-2}$$

Divide -1+i\sqrt{7} by -2.

$$x=\frac{-\sqrt{7}i+1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{-2} when ± is minus. Subtract i\sqrt{7} from -1.

$$x=\frac{-\sqrt{7}i-1}{-2}$$

Divide -1-i\sqrt{7} by -2.

$$x=\frac{1+\sqrt{7}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{7}i+1}{2} x=\frac{1+\sqrt{7}i}{2}$$

Steps for Completing the Square

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

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

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

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

To find the opposite of 2x^{2}-3x-2, find the opposite of each term.

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

Combine x^{2} and -2x^{2} to get -x^{2}.

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

Combine -2x and 3x to get x.

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

Add -3 and 2 to get -1.

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

Add 1 to both sides.

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

Add 1 and 1 to get 2.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide 2 by -1.

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

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.

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

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

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

Add -2 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=-\frac{7}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{7}i}{2} x-\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

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

$$x=\frac{1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+1}{2}$$