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

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

Solve for x (complex solution)

$$x=\frac{1+\sqrt{2}i}{2}\approx 0.5+0.707106781i$$

$$x=\frac{-\sqrt{2}i+1}{2}\approx 0.5-0.707106781i$$

Steps Using the Quadratic Formula

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

$$3x^{2}-7x+2+\left(x+1\right)\left(x+2\right)=1$$

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

$$3x^{2}-7x+2+x^{2}+3x+2=1$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-7x+2+3x+2=1$$

Combine -7x and 3x to get -4x.

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

Add 2 and 2 to get 4.

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

Subtract 1 from both sides.

$$4x^{2}-4x+4-1=0$$

Subtract 1 from 4 to get 3.

$$4x^{2}-4x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -4 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-48}}{2\times 4}$$

Add 16 to -48.

$$x=\frac{-\left(-4\right)±\sqrt{-32}}{2\times 4}$$

Take the square root of -32.

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

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{2}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{4±4\sqrt{2}i}{8}$$

Now solve the equation x=\frac{4±4\sqrt{2}i}{8} when ± is plus. Add 4 to 4i\sqrt{2}.

$$x=\frac{4+4\sqrt{2}i}{8}$$

Divide 4+4i\sqrt{2} by 8.

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

Now solve the equation x=\frac{4±4\sqrt{2}i}{8} when ± is minus. Subtract 4i\sqrt{2} from 4.

$$x=\frac{-4\sqrt{2}i+4}{8}$$

Divide 4-4i\sqrt{2} by 8.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$3x^{2}-7x+2+\left(x+1\right)\left(x+2\right)=1$$

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

$$3x^{2}-7x+2+x^{2}+3x+2=1$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-7x+2+3x+2=1$$

Combine -7x and 3x to get -4x.

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

Add 2 and 2 to get 4.

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

Subtract 4 from both sides.

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

Subtract 4 from 1 to get -3.

$$4x^{2}-4x=-3$$

Divide both sides by 4.

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

Dividing by 4 undoes the multiplication by 4.

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

Divide -4 by 4.

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

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.

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

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

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

Add -\frac{3}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

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