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

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

Solve for x (complex solution)

$$x=-\sqrt{15}i+1\approx 1-3.872983346i$$

$$x=1+\sqrt{15}i\approx 1+3.872983346i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by 2.

$$2x-4-4\left(x+3\right)=x\left(x-4\right)$$

Use the distributive property to multiply -4 by x+3.

$$2x-4-4x-12=x\left(x-4\right)$$

Combine 2x and -4x to get -2x.

$$-2x-4-12=x\left(x-4\right)$$

Subtract 12 from -4 to get -16.

$$-2x-16=x\left(x-4\right)$$

Use the distributive property to multiply x by x-4.

$$-2x-16=x^{2}-4x$$

Subtract x^{2} from both sides.

$$-2x-16-x^{2}=-4x$$

Add 4x to both sides.

$$-2x-16-x^{2}+4x=0$$

Combine -2x and 4x to get 2x.

$$2x-16-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+2x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 2.

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

Multiply -4 times -1.

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

Multiply 4 times -16.

$$x=\frac{-2±\sqrt{4-64}}{2\left(-1\right)}$$

Add 4 to -64.

$$x=\frac{-2±\sqrt{-60}}{2\left(-1\right)}$$

Take the square root of -60.

$$x=\frac{-2±2\sqrt{15}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2\sqrt{15}i}{-2}$$

Now solve the equation x=\frac{-2±2\sqrt{15}i}{-2} when ± is plus. Add -2 to 2i\sqrt{15}.

$$x=\frac{-2+2\sqrt{15}i}{-2}$$

Divide -2+2i\sqrt{15} by -2.

$$x=-\sqrt{15}i+1$$

Now solve the equation x=\frac{-2±2\sqrt{15}i}{-2} when ± is minus. Subtract 2i\sqrt{15} from -2.

$$x=\frac{-2\sqrt{15}i-2}{-2}$$

Divide -2-2i\sqrt{15} by -2.

$$x=1+\sqrt{15}i$$

The equation is now solved.

$$x=-\sqrt{15}i+1 x=1+\sqrt{15}i$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by 2.

$$2x-4-4\left(x+3\right)=x\left(x-4\right)$$

Use the distributive property to multiply -4 by x+3.

$$2x-4-4x-12=x\left(x-4\right)$$

Combine 2x and -4x to get -2x.

$$-2x-4-12=x\left(x-4\right)$$

Subtract 12 from -4 to get -16.

$$-2x-16=x\left(x-4\right)$$

Use the distributive property to multiply x by x-4.

$$-2x-16=x^{2}-4x$$

Subtract x^{2} from both sides.

$$-2x-16-x^{2}=-4x$$

Add 4x to both sides.

$$-2x-16-x^{2}+4x=0$$

Combine -2x and 4x to get 2x.

$$2x-16-x^{2}=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$2x-x^{2}=16$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+2x=16$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 2 by -1.

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

Divide 16 by -1.

$$x^{2}-2x=-16$$

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

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

Add -16 to 1.

$$x^{2}-2x+1=-15$$

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

$$\left(x-1\right)^{2}=-15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-15}$$

Simplify.

$$x-1=\sqrt{15}i x-1=-\sqrt{15}i$$

Add 1 to both sides of the equation.

$$x=1+\sqrt{15}i x=-\sqrt{15}i+1$$