Solve 3=-.006x^2+1.2x+10

$$3=-.006 { x }^{ 2 } +1.2x+10$$

Solve for x

$$x=\frac{10\sqrt{1005}}{3}+100\approx 205.672449894$$

$$x=-\frac{10\sqrt{1005}}{3}+100\approx -5.672449894$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-0.006x^{2}+1.2x+10=3$$

Subtract 3 from both sides.

$$-0.006x^{2}+1.2x+10-3=0$$

Subtract 3 from 10 to get 7.

$$-0.006x^{2}+1.2x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.006 for a, 1.2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.2±\sqrt{1.2^{2}-4\left(-0.006\right)\times 7}}{2\left(-0.006\right)}$$

Square 1.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.2±\sqrt{1.44-4\left(-0.006\right)\times 7}}{2\left(-0.006\right)}$$

Multiply -4 times -0.006.

$$x=\frac{-1.2±\sqrt{1.44+0.024\times 7}}{2\left(-0.006\right)}$$

Multiply 0.024 times 7.

$$x=\frac{-1.2±\sqrt{1.44+0.168}}{2\left(-0.006\right)}$$

Add 1.44 to 0.168 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.2±\sqrt{1.608}}{2\left(-0.006\right)}$$

Take the square root of 1.608.

$$x=\frac{-1.2±\frac{\sqrt{1005}}{25}}{2\left(-0.006\right)}$$

Multiply 2 times -0.006.

$$x=\frac{-1.2±\frac{\sqrt{1005}}{25}}{-0.012}$$

Now solve the equation x=\frac{-1.2±\frac{\sqrt{1005}}{25}}{-0.012} when ± is plus. Add -1.2 to \frac{\sqrt{1005}}{25}.

$$x=\frac{\frac{\sqrt{1005}}{25}-\frac{6}{5}}{-0.012}$$

Divide -\frac{6}{5}+\frac{\sqrt{1005}}{25} by -0.012 by multiplying -\frac{6}{5}+\frac{\sqrt{1005}}{25} by the reciprocal of -0.012.

$$x=-\frac{10\sqrt{1005}}{3}+100$$

Now solve the equation x=\frac{-1.2±\frac{\sqrt{1005}}{25}}{-0.012} when ± is minus. Subtract \frac{\sqrt{1005}}{25} from -1.2.

$$x=\frac{-\frac{\sqrt{1005}}{25}-\frac{6}{5}}{-0.012}$$

Divide -\frac{6}{5}-\frac{\sqrt{1005}}{25} by -0.012 by multiplying -\frac{6}{5}-\frac{\sqrt{1005}}{25} by the reciprocal of -0.012.

$$x=\frac{10\sqrt{1005}}{3}+100$$

The equation is now solved.

$$x=-\frac{10\sqrt{1005}}{3}+100 x=\frac{10\sqrt{1005}}{3}+100$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-0.006x^{2}+1.2x+10=3$$

Subtract 10 from both sides.

$$-0.006x^{2}+1.2x=3-10$$

Subtract 10 from 3 to get -7.

$$-0.006x^{2}+1.2x=-7$$

Divide both sides of the equation by -0.006, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.006x^{2}+1.2x}{-0.006}=-\frac{7}{-0.006}$$

Dividing by -0.006 undoes the multiplication by -0.006.

$$x^{2}+\frac{1.2}{-0.006}x=-\frac{7}{-0.006}$$

Divide 1.2 by -0.006 by multiplying 1.2 by the reciprocal of -0.006.

$$x^{2}-200x=-\frac{7}{-0.006}$$

Divide -7 by -0.006 by multiplying -7 by the reciprocal of -0.006.

$$x^{2}-200x=\frac{3500}{3}$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=\frac{3500}{3}+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=\frac{3500}{3}+10000$$

Add \frac{3500}{3} to 10000.

$$x^{2}-200x+10000=\frac{33500}{3}$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=\frac{33500}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{\frac{33500}{3}}$$

Simplify.

$$x-100=\frac{10\sqrt{1005}}{3} x-100=-\frac{10\sqrt{1005}}{3}$$

Add 100 to both sides of the equation.

$$x=\frac{10\sqrt{1005}}{3}+100 x=-\frac{10\sqrt{1005}}{3}+100$$