Solve (50-1/2m)(20m+1000)=54000

$$( 50 - \frac { 1 } { 2 } m ) ( 20 m + 1000 ) = 54000$$

Solve for m

$$m=10$$

$$m=40$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 50-\frac{1}{2}m by 20m+1000 and combine like terms.

$$500m+50000-10m^{2}=54000$$

Subtract 54000 from both sides.

$$500m+50000-10m^{2}-54000=0$$

Subtract 54000 from 50000 to get -4000.

$$500m-4000-10m^{2}=0$$

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.

$$-10m^{2}+500m-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 500 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-500±\sqrt{500^{2}-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Square 500.

$$m=\frac{-500±\sqrt{250000-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$m=\frac{-500±\sqrt{250000+40\left(-4000\right)}}{2\left(-10\right)}$$

Multiply 40 times -4000.

$$m=\frac{-500±\sqrt{250000-160000}}{2\left(-10\right)}$$

Add 250000 to -160000.

$$m=\frac{-500±\sqrt{90000}}{2\left(-10\right)}$$

Take the square root of 90000.

$$m=\frac{-500±300}{2\left(-10\right)}$$

Multiply 2 times -10.

$$m=\frac{-500±300}{-20}$$

Now solve the equation m=\frac{-500±300}{-20} when ± is plus. Add -500 to 300.

$$m=-\frac{200}{-20}$$

Divide -200 by -20.

$$m=10$$

Now solve the equation m=\frac{-500±300}{-20} when ± is minus. Subtract 300 from -500.

$$m=-\frac{800}{-20}$$

Divide -800 by -20.

$$m=40$$

The equation is now solved.

$$m=10 m=40$$

Steps for Completing the Square

Use the distributive property to multiply 50-\frac{1}{2}m by 20m+1000 and combine like terms.

$$500m+50000-10m^{2}=54000$$

Subtract 50000 from both sides.

$$500m-10m^{2}=54000-50000$$

Subtract 50000 from 54000 to get 4000.

$$500m-10m^{2}=4000$$

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.

$$-10m^{2}+500m=4000$$

Divide both sides by -10.

$$\frac{-10m^{2}+500m}{-10}=\frac{4000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$m^{2}+\frac{500}{-10}m=\frac{4000}{-10}$$

Divide 500 by -10.

$$m^{2}-50m=\frac{4000}{-10}$$

Divide 4000 by -10.

$$m^{2}-50m=-400$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-50m+\left(-25\right)^{2}=-400+\left(-25\right)^{2}$$

Square -25.

$$m^{2}-50m+625=-400+625$$

Add -400 to 625.

$$m^{2}-50m+625=225$$

Factor m^{2}-50m+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-25\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-25\right)^{2}}=\sqrt{225}$$

Simplify.

$$m-25=15 m-25=-15$$

Add 25 to both sides of the equation.

$$m=40 m=10$$