Solve (2-m)(500+1000m)+(3-m)(300+1000m)=1700

$$( 2 - m ) ( 500 + 1000 m ) + ( 3 - m ) ( 300 + 1000 m ) = 1700$$

Solve for m

$$m = \frac{\sqrt{481} + 21}{20} \approx 2.14658561$$

$$m=\frac{21-\sqrt{481}}{20}\approx -0.04658561$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2-m by 500+1000m and combine like terms.

$$1000+1500m-1000m^{2}+\left(3-m\right)\left(300+1000m\right)=1700$$

Use the distributive property to multiply 3-m by 300+1000m and combine like terms.

$$1000+1500m-1000m^{2}+900+2700m-1000m^{2}=1700$$

Add 1000 and 900 to get 1900.

$$1900+1500m-1000m^{2}+2700m-1000m^{2}=1700$$

Combine 1500m and 2700m to get 4200m.

$$1900+4200m-1000m^{2}-1000m^{2}=1700$$

Combine -1000m^{2} and -1000m^{2} to get -2000m^{2}.

$$1900+4200m-2000m^{2}=1700$$

Subtract 1700 from both sides.

$$1900+4200m-2000m^{2}-1700=0$$

Subtract 1700 from 1900 to get 200.

$$200+4200m-2000m^{2}=0$$

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.

$$-2000m^{2}+4200m+200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2000 for a, 4200 for b, and 200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-4200±\sqrt{4200^{2}-4\left(-2000\right)\times 200}}{2\left(-2000\right)}$$

Square 4200.

$$m=\frac{-4200±\sqrt{17640000-4\left(-2000\right)\times 200}}{2\left(-2000\right)}$$

Multiply -4 times -2000.

$$m=\frac{-4200±\sqrt{17640000+8000\times 200}}{2\left(-2000\right)}$$

Multiply 8000 times 200.

$$m=\frac{-4200±\sqrt{17640000+1600000}}{2\left(-2000\right)}$$

Add 17640000 to 1600000.

$$m=\frac{-4200±\sqrt{19240000}}{2\left(-2000\right)}$$

Take the square root of 19240000.

$$m=\frac{-4200±200\sqrt{481}}{2\left(-2000\right)}$$

Multiply 2 times -2000.

$$m=\frac{-4200±200\sqrt{481}}{-4000}$$

Now solve the equation m=\frac{-4200±200\sqrt{481}}{-4000} when ± is plus. Add -4200 to 200\sqrt{481}.

$$m=\frac{200\sqrt{481}-4200}{-4000}$$

Divide -4200+200\sqrt{481} by -4000.

$$m=\frac{21-\sqrt{481}}{20}$$

Now solve the equation m=\frac{-4200±200\sqrt{481}}{-4000} when ± is minus. Subtract 200\sqrt{481} from -4200.

$$m=\frac{-200\sqrt{481}-4200}{-4000}$$

Divide -4200-200\sqrt{481} by -4000.

$$m=\frac{\sqrt{481}+21}{20}$$

The equation is now solved.

$$m=\frac{21-\sqrt{481}}{20} m=\frac{\sqrt{481}+21}{20}$$

Steps for Completing the Square

Use the distributive property to multiply 2-m by 500+1000m and combine like terms.

$$1000+1500m-1000m^{2}+\left(3-m\right)\left(300+1000m\right)=1700$$

Use the distributive property to multiply 3-m by 300+1000m and combine like terms.

$$1000+1500m-1000m^{2}+900+2700m-1000m^{2}=1700$$

Add 1000 and 900 to get 1900.

$$1900+1500m-1000m^{2}+2700m-1000m^{2}=1700$$

Combine 1500m and 2700m to get 4200m.

$$1900+4200m-1000m^{2}-1000m^{2}=1700$$

Combine -1000m^{2} and -1000m^{2} to get -2000m^{2}.

$$1900+4200m-2000m^{2}=1700$$

Subtract 1900 from both sides.

$$4200m-2000m^{2}=1700-1900$$

Subtract 1900 from 1700 to get -200.

$$4200m-2000m^{2}=-200$$

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.

$$-2000m^{2}+4200m=-200$$

Divide both sides by -2000.

$$\frac{-2000m^{2}+4200m}{-2000}=-\frac{200}{-2000}$$

Dividing by -2000 undoes the multiplication by -2000.

$$m^{2}+\frac{4200}{-2000}m=-\frac{200}{-2000}$$

Reduce the fraction \frac{4200}{-2000} to lowest terms by extracting and canceling out 200.

$$m^{2}-\frac{21}{10}m=-\frac{200}{-2000}$$

Reduce the fraction \frac{-200}{-2000} to lowest terms by extracting and canceling out 200.

$$m^{2}-\frac{21}{10}m=\frac{1}{10}$$

Divide -\frac{21}{10}, the coefficient of the x term, by 2 to get -\frac{21}{20}. Then add the square of -\frac{21}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{21}{10}m+\left(-\frac{21}{20}\right)^{2}=\frac{1}{10}+\left(-\frac{21}{20}\right)^{2}$$

Square -\frac{21}{20} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{21}{10}m+\frac{441}{400}=\frac{1}{10}+\frac{441}{400}$$

Add \frac{1}{10} to \frac{441}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{21}{10}m+\frac{441}{400}=\frac{481}{400}$$

Factor m^{2}-\frac{21}{10}m+\frac{441}{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}.

$$\left(m-\frac{21}{20}\right)^{2}=\frac{481}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{21}{20}\right)^{2}}=\sqrt{\frac{481}{400}}$$

Simplify.

$$m-\frac{21}{20}=\frac{\sqrt{481}}{20} m-\frac{21}{20}=-\frac{\sqrt{481}}{20}$$

Add \frac{21}{20} to both sides of the equation.

$$m=\frac{\sqrt{481}+21}{20} m=\frac{21-\sqrt{481}}{20}$$