Solve (x-3)(x-2)=(x+4)(2-x)-14

$$(x-3)(x-2)=(x+4)(2-x)-14$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{87}i}{4}\approx 0.75+2.331844763i$$

$$x=\frac{-\sqrt{87}i+3}{4}\approx 0.75-2.331844763i$$

Steps Using the Quadratic Formula

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

$$x^{2}-5x+6=\left(x+4\right)\left(2-x\right)-14$$

Use the distributive property to multiply x+4 by 2-x and combine like terms.

$$x^{2}-5x+6=-2x-x^{2}+8-14$$

Subtract 14 from 8 to get -6.

$$x^{2}-5x+6=-2x-x^{2}-6$$

Add 2x to both sides.

$$x^{2}-5x+6+2x=-x^{2}-6$$

Combine -5x and 2x to get -3x.

$$x^{2}-3x+6=-x^{2}-6$$

Add x^{2} to both sides.

$$x^{2}-3x+6+x^{2}=-6$$

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

$$2x^{2}-3x+6=-6$$

Add 6 to both sides.

$$2x^{2}-3x+6+6=0$$

Add 6 and 6 to get 12.

$$2x^{2}-3x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\times 12}}{2\times 2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\times 12}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\times 12}}{2\times 2}$$

Multiply -8 times 12.

$$x=\frac{-\left(-3\right)±\sqrt{9-96}}{2\times 2}$$

Add 9 to -96.

$$x=\frac{-\left(-3\right)±\sqrt{-87}}{2\times 2}$$

Take the square root of -87.

$$x=\frac{-\left(-3\right)±\sqrt{87}i}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{87}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±\sqrt{87}i}{4}$$

Now solve the equation x=\frac{3±\sqrt{87}i}{4} when ± is plus. Add 3 to i\sqrt{87}.

$$x=\frac{3+\sqrt{87}i}{4}$$

Now solve the equation x=\frac{3±\sqrt{87}i}{4} when ± is minus. Subtract i\sqrt{87} from 3.

$$x=\frac{-\sqrt{87}i+3}{4}$$

The equation is now solved.

$$x=\frac{3+\sqrt{87}i}{4} x=\frac{-\sqrt{87}i+3}{4}$$

Steps for Completing the Square

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

$$x^{2}-5x+6=\left(x+4\right)\left(2-x\right)-14$$

Use the distributive property to multiply x+4 by 2-x and combine like terms.

$$x^{2}-5x+6=-2x-x^{2}+8-14$$

Subtract 14 from 8 to get -6.

$$x^{2}-5x+6=-2x-x^{2}-6$$

Add 2x to both sides.

$$x^{2}-5x+6+2x=-x^{2}-6$$

Combine -5x and 2x to get -3x.

$$x^{2}-3x+6=-x^{2}-6$$

Add x^{2} to both sides.

$$x^{2}-3x+6+x^{2}=-6$$

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

$$2x^{2}-3x+6=-6$$

Subtract 6 from both sides.

$$2x^{2}-3x=-6-6$$

Subtract 6 from -6 to get -12.

$$2x^{2}-3x=-12$$

Divide both sides by 2.

$$\frac{2x^{2}-3x}{2}=-\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{3}{2}x=-\frac{12}{2}$$

Divide -12 by 2.

$$x^{2}-\frac{3}{2}x=-6$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-6+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-6+\frac{9}{16}$$

Add -6 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{87}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=-\frac{87}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{-\frac{87}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{87}i}{4} x-\frac{3}{4}=-\frac{\sqrt{87}i}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3+\sqrt{87}i}{4} x=\frac{-\sqrt{87}i+3}{4}$$