Solve (x-10)(30-x)=144

$$( x - 10 ) ( 30 - x ) = 144$$

Solve for x (complex solution)

$$x=-2\sqrt{11}i+20\approx 20-6.633249581i$$

$$x=20+2\sqrt{11}i\approx 20+6.633249581i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by 30-x and combine like terms.

$$40x-x^{2}-300=144$$

Subtract 144 from both sides.

$$40x-x^{2}-300-144=0$$

Subtract 144 from -300 to get -444.

$$40x-x^{2}-444=0$$

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.

$$-x^{2}+40x-444=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 40 for b, and -444 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-1\right)\left(-444\right)}}{2\left(-1\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-1\right)\left(-444\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-40±\sqrt{1600+4\left(-444\right)}}{2\left(-1\right)}$$

Multiply 4 times -444.

$$x=\frac{-40±\sqrt{1600-1776}}{2\left(-1\right)}$$

Add 1600 to -1776.

$$x=\frac{-40±\sqrt{-176}}{2\left(-1\right)}$$

Take the square root of -176.

$$x=\frac{-40±4\sqrt{11}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-40±4\sqrt{11}i}{-2}$$

Now solve the equation x=\frac{-40±4\sqrt{11}i}{-2} when ± is plus. Add -40 to 4i\sqrt{11}.

$$x=\frac{-40+4\sqrt{11}i}{-2}$$

Divide -40+4i\sqrt{11} by -2.

$$x=-2\sqrt{11}i+20$$

Now solve the equation x=\frac{-40±4\sqrt{11}i}{-2} when ± is minus. Subtract 4i\sqrt{11} from -40.

$$x=\frac{-4\sqrt{11}i-40}{-2}$$

Divide -40-4i\sqrt{11} by -2.

$$x=20+2\sqrt{11}i$$

The equation is now solved.

$$x=-2\sqrt{11}i+20 x=20+2\sqrt{11}i$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by 30-x and combine like terms.

$$40x-x^{2}-300=144$$

Add 300 to both sides.

$$40x-x^{2}=144+300$$

Add 144 and 300 to get 444.

$$40x-x^{2}=444$$

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.

$$-x^{2}+40x=444$$

Divide both sides by -1.

$$\frac{-x^{2}+40x}{-1}=\frac{444}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{40}{-1}x=\frac{444}{-1}$$

Divide 40 by -1.

$$x^{2}-40x=\frac{444}{-1}$$

Divide 444 by -1.

$$x^{2}-40x=-444$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-444+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-444+400$$

Add -444 to 400.

$$x^{2}-40x+400=-44$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=-44$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{-44}$$

Simplify.

$$x-20=2\sqrt{11}i x-20=-2\sqrt{11}i$$

Add 20 to both sides of the equation.

$$x=20+2\sqrt{11}i x=-2\sqrt{11}i+20$$