Solve -0.1x^2+90x-18000=0

$$- 0.1 x ^ { 2 } + 90 x - 18000 = 0$$

Solve for x

$$x=300$$

$$x=600$$

Steps Using the Quadratic Formula

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.

$$-0.1x^{2}+90x-18000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.1 for a, 90 for b, and -18000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\left(-0.1\right)\left(-18000\right)}}{2\left(-0.1\right)}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\left(-0.1\right)\left(-18000\right)}}{2\left(-0.1\right)}$$

Multiply -4 times -0.1.

$$x=\frac{-90±\sqrt{8100+0.4\left(-18000\right)}}{2\left(-0.1\right)}$$

Multiply 0.4 times -18000.

$$x=\frac{-90±\sqrt{8100-7200}}{2\left(-0.1\right)}$$

Add 8100 to -7200.

$$x=\frac{-90±\sqrt{900}}{2\left(-0.1\right)}$$

Take the square root of 900.

$$x=\frac{-90±30}{2\left(-0.1\right)}$$

Multiply 2 times -0.1.

$$x=\frac{-90±30}{-0.2}$$

Now solve the equation x=\frac{-90±30}{-0.2} when ± is plus. Add -90 to 30.

$$x=-\frac{60}{-0.2}$$

Divide -60 by -0.2 by multiplying -60 by the reciprocal of -0.2.

$$x=300$$

Now solve the equation x=\frac{-90±30}{-0.2} when ± is minus. Subtract 30 from -90.

$$x=-\frac{120}{-0.2}$$

Divide -120 by -0.2 by multiplying -120 by the reciprocal of -0.2.

$$x=600$$

The equation is now solved.

$$x=300 x=600$$

Steps for Completing the Square

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.

$$-0.1x^{2}+90x-18000=0$$

Add 18000 to both sides of the equation.

$$-0.1x^{2}+90x-18000-\left(-18000\right)=-\left(-18000\right)$$

Subtracting -18000 from itself leaves 0.

$$-0.1x^{2}+90x=-\left(-18000\right)$$

Subtract -18000 from 0.

$$-0.1x^{2}+90x=18000$$

Multiply both sides by -10.

$$\frac{-0.1x^{2}+90x}{-0.1}=\frac{18000}{-0.1}$$

Dividing by -0.1 undoes the multiplication by -0.1.

$$x^{2}+\frac{90}{-0.1}x=\frac{18000}{-0.1}$$

Divide 90 by -0.1 by multiplying 90 by the reciprocal of -0.1.

$$x^{2}-900x=\frac{18000}{-0.1}$$

Divide 18000 by -0.1 by multiplying 18000 by the reciprocal of -0.1.

$$x^{2}-900x=-180000$$

Divide -900, the coefficient of the x term, by 2 to get -450. Then add the square of -450 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-900x+\left(-450\right)^{2}=-180000+\left(-450\right)^{2}$$

Square -450.

$$x^{2}-900x+202500=-180000+202500$$

Add -180000 to 202500.

$$x^{2}-900x+202500=22500$$

Factor x^{2}-900x+202500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-450\right)^{2}=22500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-450\right)^{2}}=\sqrt{22500}$$

Simplify.

$$x-450=150 x-450=-150$$

Add 450 to both sides of the equation.

$$x=600 x=300$$