Solve (x-4)(16-x)=48

$$(x-4)(16-x)=48$$

Solve for x (complex solution)

$$x=-2\sqrt{3}i+10\approx 10-3.464101615i$$

$$x=10+2\sqrt{3}i\approx 10+3.464101615i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-4 by 16-x and combine like terms.

$$20x-x^{2}-64=48$$

Subtract 48 from both sides.

$$20x-x^{2}-64-48=0$$

Subtract 48 from -64 to get -112.

$$20x-x^{2}-112=0$$

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.

$$-x^{2}+20x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 20 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-1\right)\left(-112\right)}}{2\left(-1\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-1\right)\left(-112\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-20±\sqrt{400+4\left(-112\right)}}{2\left(-1\right)}$$

Multiply 4 times -112.

$$x=\frac{-20±\sqrt{400-448}}{2\left(-1\right)}$$

Add 400 to -448.

$$x=\frac{-20±\sqrt{-48}}{2\left(-1\right)}$$

Take the square root of -48.

$$x=\frac{-20±4\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-20±4\sqrt{3}i}{-2}$$

Now solve the equation x=\frac{-20±4\sqrt{3}i}{-2} when ± is plus. Add -20 to 4i\sqrt{3}.

$$x=\frac{-20+4\sqrt{3}i}{-2}$$

Divide -20+4i\sqrt{3} by -2.

$$x=-2\sqrt{3}i+10$$

Now solve the equation x=\frac{-20±4\sqrt{3}i}{-2} when ± is minus. Subtract 4i\sqrt{3} from -20.

$$x=\frac{-4\sqrt{3}i-20}{-2}$$

Divide -20-4i\sqrt{3} by -2.

$$x=10+2\sqrt{3}i$$

The equation is now solved.

$$x=-2\sqrt{3}i+10 x=10+2\sqrt{3}i$$

Steps for Completing the Square

Use the distributive property to multiply x-4 by 16-x and combine like terms.

$$20x-x^{2}-64=48$$

Add 64 to both sides.

$$20x-x^{2}=48+64$$

Add 48 and 64 to get 112.

$$20x-x^{2}=112$$

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.

$$-x^{2}+20x=112$$

Divide both sides by -1.

$$\frac{-x^{2}+20x}{-1}=\frac{112}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{20}{-1}x=\frac{112}{-1}$$

Divide 20 by -1.

$$x^{2}-20x=\frac{112}{-1}$$

Divide 112 by -1.

$$x^{2}-20x=-112$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-112+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-112+100$$

Add -112 to 100.

$$x^{2}-20x+100=-12$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=-12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{-12}$$

Simplify.

$$x-10=2\sqrt{3}i x-10=-2\sqrt{3}i$$

Add 10 to both sides of the equation.

$$x=10+2\sqrt{3}i x=-2\sqrt{3}i+10$$