Solve (40-x)(20+2x)=1500

$$(40-x)(20+2x)=1500$$

Solve for x (complex solution)

$$x=15+5\sqrt{5}i\approx 15+11.180339887i$$

$$x=-5\sqrt{5}i+15\approx 15-11.180339887i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-x by 20+2x and combine like terms.

$$800+60x-2x^{2}=1500$$

Subtract 1500 from both sides.

$$800+60x-2x^{2}-1500=0$$

Subtract 1500 from 800 to get -700.

$$-700+60x-2x^{2}=0$$

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.

$$-2x^{2}+60x-700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 60 for b, and -700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-60±\sqrt{3600+8\left(-700\right)}}{2\left(-2\right)}$$

Multiply 8 times -700.

$$x=\frac{-60±\sqrt{3600-5600}}{2\left(-2\right)}$$

Add 3600 to -5600.

$$x=\frac{-60±\sqrt{-2000}}{2\left(-2\right)}$$

Take the square root of -2000.

$$x=\frac{-60±20\sqrt{5}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-60±20\sqrt{5}i}{-4}$$

Now solve the equation x=\frac{-60±20\sqrt{5}i}{-4} when ± is plus. Add -60 to 20i\sqrt{5}.

$$x=\frac{-60+20\sqrt{5}i}{-4}$$

Divide -60+20i\sqrt{5} by -4.

$$x=-5\sqrt{5}i+15$$

Now solve the equation x=\frac{-60±20\sqrt{5}i}{-4} when ± is minus. Subtract 20i\sqrt{5} from -60.

$$x=\frac{-20\sqrt{5}i-60}{-4}$$

Divide -60-20i\sqrt{5} by -4.

$$x=15+5\sqrt{5}i$$

The equation is now solved.

$$x=-5\sqrt{5}i+15 x=15+5\sqrt{5}i$$

Steps for Completing the Square

Use the distributive property to multiply 40-x by 20+2x and combine like terms.

$$800+60x-2x^{2}=1500$$

Subtract 800 from both sides.

$$60x-2x^{2}=1500-800$$

Subtract 800 from 1500 to get 700.

$$60x-2x^{2}=700$$

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.

$$-2x^{2}+60x=700$$

Divide both sides by -2.

$$\frac{-2x^{2}+60x}{-2}=\frac{700}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{60}{-2}x=\frac{700}{-2}$$

Divide 60 by -2.

$$x^{2}-30x=\frac{700}{-2}$$

Divide 700 by -2.

$$x^{2}-30x=-350$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-350+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-350+225$$

Add -350 to 225.

$$x^{2}-30x+225=-125$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=-125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{-125}$$

Simplify.

$$x-15=5\sqrt{5}i x-15=-5\sqrt{5}i$$

Add 15 to both sides of the equation.

$$x=15+5\sqrt{5}i x=-5\sqrt{5}i+15$$