Solve (1+x/2)(1000-200x)+500(1+x)=14400

$$(1+ \frac{ x }{ 2 } )(1000-200x)+500(1+x)=14400$$

Solve for x (complex solution)

$$x=4+\sqrt{113}i\approx 4+10.630145813i$$

$$x=-\sqrt{113}i+4\approx 4-10.630145813i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\left(1+\frac{x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Use the distributive property to multiply 2 by 1+\frac{x}{2}.

$$\left(2+2\times \frac{x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Express 2\times \frac{x}{2} as a single fraction.

$$\left(2+\frac{2x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Cancel out 2 and 2.

$$\left(2+x\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Apply the distributive property by multiplying each term of 2+x by each term of 1000-200x.

$$2000-400x+1000x-200x^{2}+1000\left(1+x\right)=28800$$

Combine -400x and 1000x to get 600x.

$$2000+600x-200x^{2}+1000\left(1+x\right)=28800$$

Use the distributive property to multiply 1000 by 1+x.

$$2000+600x-200x^{2}+1000+1000x=28800$$

Add 2000 and 1000 to get 3000.

$$3000+600x-200x^{2}+1000x=28800$$

Combine 600x and 1000x to get 1600x.

$$3000+1600x-200x^{2}=28800$$

Subtract 28800 from both sides.

$$3000+1600x-200x^{2}-28800=0$$

Subtract 28800 from 3000 to get -25800.

$$-25800+1600x-200x^{2}=0$$

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.

$$-200x^{2}+1600x-25800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -200 for a, 1600 for b, and -25800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1600±\sqrt{1600^{2}-4\left(-200\right)\left(-25800\right)}}{2\left(-200\right)}$$

Square 1600.

$$x=\frac{-1600±\sqrt{2560000-4\left(-200\right)\left(-25800\right)}}{2\left(-200\right)}$$

Multiply -4 times -200.

$$x=\frac{-1600±\sqrt{2560000+800\left(-25800\right)}}{2\left(-200\right)}$$

Multiply 800 times -25800.

$$x=\frac{-1600±\sqrt{2560000-20640000}}{2\left(-200\right)}$$

Add 2560000 to -20640000.

$$x=\frac{-1600±\sqrt{-18080000}}{2\left(-200\right)}$$

Take the square root of -18080000.

$$x=\frac{-1600±400\sqrt{113}i}{2\left(-200\right)}$$

Multiply 2 times -200.

$$x=\frac{-1600±400\sqrt{113}i}{-400}$$

Now solve the equation x=\frac{-1600±400\sqrt{113}i}{-400} when ± is plus. Add -1600 to 400i\sqrt{113}.

$$x=\frac{-1600+400\sqrt{113}i}{-400}$$

Divide -1600+400i\sqrt{113} by -400.

$$x=-\sqrt{113}i+4$$

Now solve the equation x=\frac{-1600±400\sqrt{113}i}{-400} when ± is minus. Subtract 400i\sqrt{113} from -1600.

$$x=\frac{-400\sqrt{113}i-1600}{-400}$$

Divide -1600-400i\sqrt{113} by -400.

$$x=4+\sqrt{113}i$$

The equation is now solved.

$$x=-\sqrt{113}i+4 x=4+\sqrt{113}i$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\left(1+\frac{x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Use the distributive property to multiply 2 by 1+\frac{x}{2}.

$$\left(2+2\times \frac{x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Express 2\times \frac{x}{2} as a single fraction.

$$\left(2+\frac{2x}{2}\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Cancel out 2 and 2.

$$\left(2+x\right)\left(1000-200x\right)+1000\left(1+x\right)=28800$$

Apply the distributive property by multiplying each term of 2+x by each term of 1000-200x.

$$2000-400x+1000x-200x^{2}+1000\left(1+x\right)=28800$$

Combine -400x and 1000x to get 600x.

$$2000+600x-200x^{2}+1000\left(1+x\right)=28800$$

Use the distributive property to multiply 1000 by 1+x.

$$2000+600x-200x^{2}+1000+1000x=28800$$

Add 2000 and 1000 to get 3000.

$$3000+600x-200x^{2}+1000x=28800$$

Combine 600x and 1000x to get 1600x.

$$3000+1600x-200x^{2}=28800$$

Subtract 3000 from both sides.

$$1600x-200x^{2}=28800-3000$$

Subtract 3000 from 28800 to get 25800.

$$1600x-200x^{2}=25800$$

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.

$$-200x^{2}+1600x=25800$$

Divide both sides by -200.

$$\frac{-200x^{2}+1600x}{-200}=\frac{25800}{-200}$$

Dividing by -200 undoes the multiplication by -200.

$$x^{2}+\frac{1600}{-200}x=\frac{25800}{-200}$$

Divide 1600 by -200.

$$x^{2}-8x=\frac{25800}{-200}$$

Divide 25800 by -200.

$$x^{2}-8x=-129$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-129+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-129+16$$

Add -129 to 16.

$$x^{2}-8x+16=-113$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=-113$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-113}$$

Simplify.

$$x-4=\sqrt{113}i x-4=-\sqrt{113}i$$

Add 4 to both sides of the equation.

$$x=4+\sqrt{113}i x=-\sqrt{113}i+4$$