Solve (x+40)(50-x/2)=2250

$$( x + 40 ) ( 50 - \frac { x } { 2 } ) = 2250$$

Solve for x

$$x=10$$

$$x=50$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\left(x+40\right)\left(50-\frac{x}{2}\right)=4500$$

Use the distributive property to multiply 2 by x+40.

$$\left(2x+80\right)\left(50-\frac{x}{2}\right)=4500$$

Apply the distributive property by multiplying each term of 2x+80 by each term of 50-\frac{x}{2}.

$$100x+2x\left(-\frac{x}{2}\right)+4000+80\left(-\frac{x}{2}\right)=4500$$

Express 2\left(-\frac{x}{2}\right) as a single fraction.

$$100x+\frac{-2x}{2}x+4000+80\left(-\frac{x}{2}\right)=4500$$

Cancel out 2 and 2.

$$100x-xx+4000+80\left(-\frac{x}{2}\right)=4500$$

Cancel out 2, the greatest common factor in 80 and 2.

$$100x-xx+4000-40x=4500$$

Combine 100x and -40x to get 60x.

$$60x-xx+4000=4500$$

Multiply x and x to get x^{2}.

$$60x-x^{2}+4000=4500$$

Subtract 4500 from both sides.

$$60x-x^{2}[phone]=0$$

Subtract 4500 from 4000 to get -500.

$$60x-x^{2}-500=0$$

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.

$$-x^{2}+60x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 60 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-60±\sqrt{3600+4\left(-500\right)}}{2\left(-1\right)}$$

Multiply 4 times -500.

$$x=\frac{-60±\sqrt{3600-2000}}{2\left(-1\right)}$$

Add 3600 to -2000.

$$x=\frac{-60±\sqrt{1600}}{2\left(-1\right)}$$

Take the square root of 1600.

$$x=\frac{-60±40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-60±40}{-2}$$

Now solve the equation x=\frac{-60±40}{-2} when ± is plus. Add -60 to 40.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

Now solve the equation x=\frac{-60±40}{-2} when ± is minus. Subtract 40 from -60.

$$x=-\frac{100}{-2}$$

Divide -100 by -2.

$$x=50$$

The equation is now solved.

$$x=10 x=50$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\left(x+40\right)\left(50-\frac{x}{2}\right)=4500$$

Use the distributive property to multiply 2 by x+40.

$$\left(2x+80\right)\left(50-\frac{x}{2}\right)=4500$$

Apply the distributive property by multiplying each term of 2x+80 by each term of 50-\frac{x}{2}.

$$100x+2x\left(-\frac{x}{2}\right)+4000+80\left(-\frac{x}{2}\right)=4500$$

Express 2\left(-\frac{x}{2}\right) as a single fraction.

$$100x+\frac{-2x}{2}x+4000+80\left(-\frac{x}{2}\right)=4500$$

Cancel out 2 and 2.

$$100x-xx+4000+80\left(-\frac{x}{2}\right)=4500$$

Cancel out 2, the greatest common factor in 80 and 2.

$$100x-xx+4000-40x=4500$$

Combine 100x and -40x to get 60x.

$$60x-xx+4000=4500$$

Multiply x and x to get x^{2}.

$$60x-x^{2}+4000=4500$$

Subtract 4000 from both sides.

$$60x-x^{2}=4500-4000$$

Subtract 4000 from 4500 to get 500.

$$60x-x^{2}=500$$

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.

$$-x^{2}+60x=500$$

Divide both sides by -1.

$$\frac{-x^{2}+60x}{-1}=\frac{500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{60}{-1}x=\frac{500}{-1}$$

Divide 60 by -1.

$$x^{2}-60x=\frac{500}{-1}$$

Divide 500 by -1.

$$x^{2}-60x=-500$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-500+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-500+900$$

Add -500 to 900.

$$x^{2}-60x+900=400$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-30=20 x-30=-20$$

Add 30 to both sides of the equation.

$$x=50 x=10$$