Solve (12+x)(180-10x)=2000

$$(12+x)(180-10x)=2000$$

Solve for x

$$x=-2$$

$$x=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12+x by 180-10x and combine like terms.

$$2160+60x-10x^{2}=2000$$

Subtract 2000 from both sides.

$$2160+60x-10x^{2}-2000=0$$

Subtract 2000 from 2160 to get 160.

$$160+60x-10x^{2}=0$$

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.

$$-10x^{2}+60x+160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 60 for b, and 160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-10\right)\times 160}}{2\left(-10\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-10\right)\times 160}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-60±\sqrt{3600+40\times 160}}{2\left(-10\right)}$$

Multiply 40 times 160.

$$x=\frac{-60±\sqrt{3600+6400}}{2\left(-10\right)}$$

Add 3600 to 6400.

$$x=\frac{-60±\sqrt{10000}}{2\left(-10\right)}$$

Take the square root of 10000.

$$x=\frac{-60±100}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-60±100}{-20}$$

Now solve the equation x=\frac{-60±100}{-20} when ± is plus. Add -60 to 100.

$$x=\frac{40}{-20}$$

Divide 40 by -20.

$$x=-2$$

Now solve the equation x=\frac{-60±100}{-20} when ± is minus. Subtract 100 from -60.

$$x=-\frac{160}{-20}$$

Divide -160 by -20.

$$x=8$$

The equation is now solved.

$$x=-2 x=8$$

Steps for Completing the Square

Use the distributive property to multiply 12+x by 180-10x and combine like terms.

$$2160+60x-10x^{2}=2000$$

Subtract 2160 from both sides.

$$60x-10x^{2}=2000-2160$$

Subtract 2160 from 2000 to get -160.

$$60x-10x^{2}=-160$$

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.

$$-10x^{2}+60x=-160$$

Divide both sides by -10.

$$\frac{-10x^{2}+60x}{-10}=-\frac{160}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{60}{-10}x=-\frac{160}{-10}$$

Divide 60 by -10.

$$x^{2}-6x=-\frac{160}{-10}$$

Divide -160 by -10.

$$x^{2}-6x=16$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=16+9$$

Add 16 to 9.

$$x^{2}-6x+9=25$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-3=5 x-3=-5$$

Add 3 to both sides of the equation.

$$x=8 x=-2$$