Solve (200-x)(20+2x)=1750

$$(200-x)(20+2x)=1750$$

Solve for x

$$x=5\sqrt{406}+95\approx 195.747208398$$

$$x=95-5\sqrt{406}\approx -5.747208398$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 200-x by 20+2x and combine like terms.

$$4000+380x-2x^{2}=1750$$

Subtract 1750 from both sides.

$$4000+380x-2x^{2}-1750=0$$

Subtract 1750 from 4000 to get 2250.

$$2250+380x-2x^{2}=0$$

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.

$$-2x^{2}+380x+2250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 380 for b, and 2250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-380±\sqrt{380^{2}-4\left(-2\right)\times 2250}}{2\left(-2\right)}$$

Square 380.

$$x=\frac{-380±\sqrt{144400-4\left(-2\right)\times 2250}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-380±\sqrt{144400+8\times 2250}}{2\left(-2\right)}$$

Multiply 8 times 2250.

$$x=\frac{-380±\sqrt{144400+18000}}{2\left(-2\right)}$$

Add 144400 to 18000.

$$x=\frac{-380±\sqrt{162400}}{2\left(-2\right)}$$

Take the square root of 162400.

$$x=\frac{-380±20\sqrt{406}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-380±20\sqrt{406}}{-4}$$

Now solve the equation x=\frac{-380±20\sqrt{406}}{-4} when ± is plus. Add -380 to 20\sqrt{406}.

$$x=\frac{20\sqrt{406}-380}{-4}$$

Divide -380+20\sqrt{406} by -4.

$$x=95-5\sqrt{406}$$

Now solve the equation x=\frac{-380±20\sqrt{406}}{-4} when ± is minus. Subtract 20\sqrt{406} from -380.

$$x=\frac{-20\sqrt{406}-380}{-4}$$

Divide -380-20\sqrt{406} by -4.

$$x=5\sqrt{406}+95$$

The equation is now solved.

$$x=95-5\sqrt{406} x=5\sqrt{406}+95$$

Steps for Completing the Square

Use the distributive property to multiply 200-x by 20+2x and combine like terms.

$$4000+380x-2x^{2}=1750$$

Subtract 4000 from both sides.

$$380x-2x^{2}=1750-4000$$

Subtract 4000 from 1750 to get -2250.

$$380x-2x^{2}=-2250$$

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.

$$-2x^{2}+380x=-2250$$

Divide both sides by -2.

$$\frac{-2x^{2}+380x}{-2}=-\frac{2250}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{380}{-2}x=-\frac{2250}{-2}$$

Divide 380 by -2.

$$x^{2}-190x=-\frac{2250}{-2}$$

Divide -2250 by -2.

$$x^{2}-190x=1125$$

Divide -190, the coefficient of the x term, by 2 to get -95. Then add the square of -95 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-190x+\left(-95\right)^{2}=1125+\left(-95\right)^{2}$$

Square -95.

$$x^{2}-190x+9025=1125+9025$$

Add 1125 to 9025.

$$x^{2}-190x+9025=10150$$

Factor x^{2}-190x+9025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-95\right)^{2}=10150$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-95\right)^{2}}=\sqrt{10150}$$

Simplify.

$$x-95=5\sqrt{406} x-95=-5\sqrt{406}$$

Add 95 to both sides of the equation.

$$x=5\sqrt{406}+95 x=95-5\sqrt{406}$$