Solve (40+x)(50-1/2x)=2250

$$( 40 + x ) ( 50 - \frac { 1 } { 2 } x ) = 2250$$

Solve for x

$$x=10$$

$$x=50$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40+x by 50-\frac{1}{2}x and combine like terms.

$$2000+30x-\frac{1}{2}x^{2}=2250$$

Subtract 2250 from both sides.

$$2000+30x-\frac{1}{2}x^{2}-2250=0$$

Subtract 2250 from 2000 to get -250.

$$-250+30x-\frac{1}{2}x^{2}=0$$

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.

$$-\frac{1}{2}x^{2}+30x-250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, 30 for b, and -250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-\frac{1}{2}\right)\left(-250\right)}}{2\left(-\frac{1}{2}\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-\frac{1}{2}\right)\left(-250\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply -4 times -\frac{1}{2}.

$$x=\frac{-30±\sqrt{900+2\left(-250\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -250.

$$x=\frac{-30±\sqrt{900-500}}{2\left(-\frac{1}{2}\right)}$$

Add 900 to -500.

$$x=\frac{-30±\sqrt{400}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of 400.

$$x=\frac{-30±20}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -\frac{1}{2}.

$$x=\frac{-30±20}{-1}$$

Now solve the equation x=\frac{-30±20}{-1} when ± is plus. Add -30 to 20.

$$x=-\frac{10}{-1}$$

Divide -10 by -1.

$$x=10$$

Now solve the equation x=\frac{-30±20}{-1} when ± is minus. Subtract 20 from -30.

$$x=-\frac{50}{-1}$$

Divide -50 by -1.

$$x=50$$

The equation is now solved.

$$x=10 x=50$$

Steps for Completing the Square

Use the distributive property to multiply 40+x by 50-\frac{1}{2}x and combine like terms.

$$2000+30x-\frac{1}{2}x^{2}=2250$$

Subtract 2000 from both sides.

$$30x-\frac{1}{2}x^{2}=2250-2000$$

Subtract 2000 from 2250 to get 250.

$$30x-\frac{1}{2}x^{2}=250$$

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.

$$-\frac{1}{2}x^{2}+30x=250$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}+30x}{-\frac{1}{2}}=\frac{250}{-\frac{1}{2}}$$

Dividing by -\frac{1}{2} undoes the multiplication by -\frac{1}{2}.

$$x^{2}+\frac{30}{-\frac{1}{2}}x=\frac{250}{-\frac{1}{2}}$$

Divide 30 by -\frac{1}{2} by multiplying 30 by the reciprocal of -\frac{1}{2}.

$$x^{2}-60x=\frac{250}{-\frac{1}{2}}$$

Divide 250 by -\frac{1}{2} by multiplying 250 by the reciprocal of -\frac{1}{2}.

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

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.

$$x=50 x=10$$