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

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

Solve for m

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

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

Steps Using the Quadratic Formula

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

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

Subtract 300 from 800 to get 500.

$$1000+300m-100m^{2}+\left(1+m\right)\times 500=14400$$

Use the distributive property to multiply 1+m by 500.

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

Add 1000 and 500 to get 1500.

$$1500+300m-100m^{2}+500m=14400$$

Combine 300m and 500m to get 800m.

$$1500+800m-100m^{2}=14400$$

Subtract 14400 from both sides.

$$1500+800m-100m^{2}-14400=0$$

Subtract 14400 from 1500 to get -12900.

$$-12900+800m-100m^{2}=0$$

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.

$$-100m^{2}+800m-12900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 800 for b, and -12900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-800±\sqrt{800^{2}-4\left(-100\right)\left(-12900\right)}}{2\left(-100\right)}$$

Square 800.

$$m=\frac{-800±\sqrt{640000-4\left(-100\right)\left(-12900\right)}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$m=\frac{-800±\sqrt{640000+400\left(-12900\right)}}{2\left(-100\right)}$$

Multiply 400 times -12900.

$$m=\frac{-800±\sqrt{640000-5160000}}{2\left(-100\right)}$$

Add 640000 to -5160000.

$$m=\frac{-800±\sqrt{-4520000}}{2\left(-100\right)}$$

Take the square root of -4520000.

$$m=\frac{-800±200\sqrt{113}i}{2\left(-100\right)}$$

Multiply 2 times -100.

$$m=\frac{-800±200\sqrt{113}i}{-200}$$

Now solve the equation m=\frac{-800±200\sqrt{113}i}{-200} when ± is plus. Add -800 to 200i\sqrt{113}.

$$m=\frac{-800+200\sqrt{113}i}{-200}$$

Divide -800+200i\sqrt{113} by -200.

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

Now solve the equation m=\frac{-800±200\sqrt{113}i}{-200} when ± is minus. Subtract 200i\sqrt{113} from -800.

$$m=\frac{-200\sqrt{113}i-800}{-200}$$

Divide -800-200i\sqrt{113} by -200.

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

The equation is now solved.

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

Steps for Completing the Square

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

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

Subtract 300 from 800 to get 500.

$$1000+300m-100m^{2}+\left(1+m\right)\times 500=14400$$

Use the distributive property to multiply 1+m by 500.

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

Add 1000 and 500 to get 1500.

$$1500+300m-100m^{2}+500m=14400$$

Combine 300m and 500m to get 800m.

$$1500+800m-100m^{2}=14400$$

Subtract 1500 from both sides.

$$800m-100m^{2}=14400-1500$$

Subtract 1500 from 14400 to get 12900.

$$800m-100m^{2}=12900$$

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.

$$-100m^{2}+800m=12900$$

Divide both sides by -100.

$$\frac{-100m^{2}+800m}{-100}=\frac{12900}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$m^{2}+\frac{800}{-100}m=\frac{12900}{-100}$$

Divide 800 by -100.

$$m^{2}-8m=\frac{12900}{-100}$$

Divide 12900 by -100.

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

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.

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

Square -4.

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

Add -129 to 16.

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

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

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 4 to both sides of the equation.

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