Solve (30-20-x)(200-10x)=2160

$$( 30 - 20 - x ) ( 200 - 10 x ) = 2160$$

Solve for x

$$x=\sqrt{241}+15\approx 30.524174696$$

$$x=15-\sqrt{241}\approx -0.524174696$$

Steps Using the Quadratic Formula

Subtract 20 from 30 to get 10.

$$\left(10-x\right)\left(200-10x\right)=2160$$

Use the distributive property to multiply 10-x by 200-10x and combine like terms.

$$2000-300x+10x^{2}=2160$$

Subtract 2160 from both sides.

$$2000-300x+10x^{2}-2160=0$$

Subtract 2160 from 2000 to get -160.

$$-160-300x+10x^{2}=0$$

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.

$$10x^{2}-300x-160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -300 for b, and -160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\times 10\left(-160\right)}}{2\times 10}$$

Square -300.

$$x=\frac{-\left(-300\right)±\sqrt{90000-4\times 10\left(-160\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-300\right)±\sqrt{90000-40\left(-160\right)}}{2\times 10}$$

Multiply -40 times -160.

$$x=\frac{-\left(-300\right)±\sqrt{90000+6400}}{2\times 10}$$

Add 90000 to 6400.

$$x=\frac{-\left(-300\right)±\sqrt{96400}}{2\times 10}$$

Take the square root of 96400.

$$x=\frac{-\left(-300\right)±20\sqrt{241}}{2\times 10}$$

The opposite of -300 is 300.

$$x=\frac{300±20\sqrt{241}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{300±20\sqrt{241}}{20}$$

Now solve the equation x=\frac{300±20\sqrt{241}}{20} when ± is plus. Add 300 to 20\sqrt{241}.

$$x=\frac{20\sqrt{241}+300}{20}$$

Divide 300+20\sqrt{241} by 20.

$$x=\sqrt{241}+15$$

Now solve the equation x=\frac{300±20\sqrt{241}}{20} when ± is minus. Subtract 20\sqrt{241} from 300.

$$x=\frac{300-20\sqrt{241}}{20}$$

Divide 300-20\sqrt{241} by 20.

$$x=15-\sqrt{241}$$

The equation is now solved.

$$x=\sqrt{241}+15 x=15-\sqrt{241}$$

Steps for Completing the Square

Subtract 20 from 30 to get 10.

$$\left(10-x\right)\left(200-10x\right)=2160$$

Use the distributive property to multiply 10-x by 200-10x and combine like terms.

$$2000-300x+10x^{2}=2160$$

Subtract 2000 from both sides.

$$-300x+10x^{2}=2160-2000$$

Subtract 2000 from 2160 to get 160.

$$-300x+10x^{2}=160$$

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.

$$10x^{2}-300x=160$$

Divide both sides by 10.

$$\frac{10x^{2}-300x}{10}=\frac{160}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{300}{10}\right)x=\frac{160}{10}$$

Divide -300 by 10.

$$x^{2}-30x=\frac{160}{10}$$

Divide 160 by 10.

$$x^{2}-30x=16$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=16+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=16+225$$

Add 16 to 225.

$$x^{2}-30x+225=241$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=241$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{241}$$

Simplify.

$$x-15=\sqrt{241} x-15=-\sqrt{241}$$

Add 15 to both sides of the equation.

$$x=\sqrt{241}+15 x=15-\sqrt{241}$$