Solve 0.2m^2=-0.3m+0.1

$$0.2 m ^ { 2 } = - 0.3 m + 0.1$$

Solve for m

$$m=\frac{\sqrt{17}-3}{4}\approx 0.280776406$$

$$m=\frac{-\sqrt{17}-3}{4}\approx -1.780776406$$

Steps Using the Quadratic Formula

Add 0.3m to both sides.

$$0.2m^{2}+0.3m=0.1$$

Subtract 0.1 from both sides.

$$0.2m^{2}+0.3m-0.1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.2 for a, 0.3 for b, and -0.1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-0.3±\sqrt{0.3^{2}-4\times 0.2\left(-0.1\right)}}{2\times 0.2}$$

Square 0.3 by squaring both the numerator and the denominator of the fraction.

$$m=\frac{-0.3±\sqrt{0.09-4\times 0.2\left(-0.1\right)}}{2\times 0.2}$$

Multiply -4 times 0.2.

$$m=\frac{-0.3±\sqrt{0.09-0.8\left(-0.1\right)}}{2\times 0.2}$$

Multiply -0.8 times -0.1 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$m=\frac{-0.3±\sqrt{0.09+0.08}}{2\times 0.2}$$

Add 0.09 to 0.08 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{-0.3±\sqrt{0.17}}{2\times 0.2}$$

Take the square root of 0.17.

$$m=\frac{-0.3±\frac{\sqrt{17}}{10}}{2\times 0.2}$$

Multiply 2 times 0.2.

$$m=\frac{-0.3±\frac{\sqrt{17}}{10}}{0.4}$$

Now solve the equation m=\frac{-0.3±\frac{\sqrt{17}}{10}}{0.4} when ± is plus. Add -0.3 to \frac{\sqrt{17}}{10}.

$$m=\frac{\sqrt{17}-3}{0.4\times 10}$$

Divide \frac{-3+\sqrt{17}}{10} by 0.4 by multiplying \frac{-3+\sqrt{17}}{10} by the reciprocal of 0.4.

$$m=\frac{\sqrt{17}-3}{4}$$

Now solve the equation m=\frac{-0.3±\frac{\sqrt{17}}{10}}{0.4} when ± is minus. Subtract \frac{\sqrt{17}}{10} from -0.3.

$$m=\frac{-\sqrt{17}-3}{0.4\times 10}$$

Divide \frac{-3-\sqrt{17}}{10} by 0.4 by multiplying \frac{-3-\sqrt{17}}{10} by the reciprocal of 0.4.

$$m=\frac{-\sqrt{17}-3}{4}$$

The equation is now solved.

$$m=\frac{\sqrt{17}-3}{4} m=\frac{-\sqrt{17}-3}{4}$$

Steps for Completing the Square

Add 0.3m to both sides.

$$0.2m^{2}+0.3m=0.1$$

Multiply both sides by 5.

$$\frac{0.2m^{2}+0.3m}{0.2}=\frac{0.1}{0.2}$$

Dividing by 0.2 undoes the multiplication by 0.2.

$$m^{2}+\frac{0.3}{0.2}m=\frac{0.1}{0.2}$$

Divide 0.3 by 0.2 by multiplying 0.3 by the reciprocal of 0.2.

$$m^{2}+1.5m=\frac{0.1}{0.2}$$

Divide 0.1 by 0.2 by multiplying 0.1 by the reciprocal of 0.2.

$$m^{2}+1.5m=0.5$$

Divide 1.5, the coefficient of the x term, by 2 to get 0.75. Then add the square of 0.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+1.5m+0.75^{2}=0.5+0.75^{2}$$

Square 0.75 by squaring both the numerator and the denominator of the fraction.

$$m^{2}+1.5m+0.5625=0.5+0.5625$$

Add 0.5 to 0.5625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+1.5m+0.5625=1.0625$$

Factor m^{2}+1.5m+0.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+0.75\right)^{2}=1.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+0.75\right)^{2}}=\sqrt{1.0625}$$

Simplify.

$$m+0.75=\frac{\sqrt{17}}{4} m+0.75=-\frac{\sqrt{17}}{4}$$

Subtract 0.75 from both sides of the equation.

$$m=\frac{\sqrt{17}-3}{4} m=\frac{-\sqrt{17}-3}{4}$$