Solve left(1+1/2xright)left(1000-200xright)+left(1+xright)left(800-300xright)=14400

$$\left( 1+ \frac{ 1 }{ 2 } x \right) \left( 1000-200x \right) + \left( 1+x \right) \left( 800-300x \right) = 14400$$

Solve for x (complex solution)

$$x=\frac{\sqrt{122}i}{2}+1\approx 1+5.522680509i$$

$$x=-\frac{\sqrt{122}i}{2}+1\approx 1-5.522680509i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1+\frac{1}{2}x by 1000-200x and combine like terms.

$$1000+300x-100x^{2}+\left(1+x\right)\left(800-300x\right)=14400$$

Use the distributive property to multiply 1+x by 800-300x and combine like terms.

$$1000+300x-100x^{2}+800+500x-300x^{2}=14400$$

Add 1000 and 800 to get 1800.

$$1800+300x-100x^{2}+500x-300x^{2}=14400$$

Combine 300x and 500x to get 800x.

$$1800+800x-100x^{2}-300x^{2}=14400$$

Combine -100x^{2} and -300x^{2} to get -400x^{2}.

$$1800+800x-400x^{2}=14400$$

Subtract 14400 from both sides.

$$1800+800x-400x^{2}-14400=0$$

Subtract 14400 from 1800 to get -12600.

$$-12600+800x-400x^{2}=0$$

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.

$$-400x^{2}+800x-12600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -400 for a, 800 for b, and -12600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-800±\sqrt{800^{2}-4\left(-400\right)\left(-12600\right)}}{2\left(-400\right)}$$

Square 800.

$$x=\frac{-800±\sqrt{640000-4\left(-400\right)\left(-12600\right)}}{2\left(-400\right)}$$

Multiply -4 times -400.

$$x=\frac{-800±\sqrt{640000+1600\left(-12600\right)}}{2\left(-400\right)}$$

Multiply 1600 times -12600.

$$x=\frac{-800±\sqrt{640000-20160000}}{2\left(-400\right)}$$

Add 640000 to -20160000.

$$x=\frac{-800±\sqrt{-19520000}}{2\left(-400\right)}$$

Take the square root of -19520000.

$$x=\frac{-800±400\sqrt{122}i}{2\left(-400\right)}$$

Multiply 2 times -400.

$$x=\frac{-800±400\sqrt{122}i}{-800}$$

Now solve the equation x=\frac{-800±400\sqrt{122}i}{-800} when ± is plus. Add -800 to 400i\sqrt{122}.

$$x=\frac{-800+400\sqrt{122}i}{-800}$$

Divide -800+400i\sqrt{122} by -800.

$$x=-\frac{\sqrt{122}i}{2}+1$$

Now solve the equation x=\frac{-800±400\sqrt{122}i}{-800} when ± is minus. Subtract 400i\sqrt{122} from -800.

$$x=\frac{-400\sqrt{122}i-800}{-800}$$

Divide -800-400i\sqrt{122} by -800.

$$x=\frac{\sqrt{122}i}{2}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{122}i}{2}+1 x=\frac{\sqrt{122}i}{2}+1$$

Steps for Completing the Square

Use the distributive property to multiply 1+\frac{1}{2}x by 1000-200x and combine like terms.

$$1000+300x-100x^{2}+\left(1+x\right)\left(800-300x\right)=14400$$

Use the distributive property to multiply 1+x by 800-300x and combine like terms.

$$1000+300x-100x^{2}+800+500x-300x^{2}=14400$$

Add 1000 and 800 to get 1800.

$$1800+300x-100x^{2}+500x-300x^{2}=14400$$

Combine 300x and 500x to get 800x.

$$1800+800x-100x^{2}-300x^{2}=14400$$

Combine -100x^{2} and -300x^{2} to get -400x^{2}.

$$1800+800x-400x^{2}=14400$$

Subtract 1800 from both sides.

$$800x-400x^{2}=14400-1800$$

Subtract 1800 from 14400 to get 12600.

$$800x-400x^{2}=12600$$

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.

$$-400x^{2}+800x=12600$$

Divide both sides by -400.

$$\frac{-400x^{2}+800x}{-400}=\frac{12600}{-400}$$

Dividing by -400 undoes the multiplication by -400.

$$x^{2}+\frac{800}{-400}x=\frac{12600}{-400}$$

Divide 800 by -400.

$$x^{2}-2x=\frac{12600}{-400}$$

Reduce the fraction \frac{12600}{-400} to lowest terms by extracting and canceling out 200.

$$x^{2}-2x=-\frac{63}{2}$$

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.

$$x^{2}-2x+1=-\frac{63}{2}+1$$

Add -\frac{63}{2} to 1.

$$x^{2}-2x+1=-\frac{61}{2}$$

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}.

$$\left(x-1\right)^{2}=-\frac{61}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{61}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{122}i}{2} x-1=-\frac{\sqrt{122}i}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{122}i}{2}+1 x=-\frac{\sqrt{122}i}{2}+1$$