Solve 0=14m^2+21m+1

$$0 = 14 m ^ { 2 } + 21 m + 1$$

Solve for m

$$m=\frac{\sqrt{385}}{28}-\frac{3}{4}\approx -0.049235112$$

$$m=-\frac{\sqrt{385}}{28}-\frac{3}{4}\approx -1.450764888$$

Steps Using the Quadratic Formula

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

$$14m^{2}+21m+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, 21 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-21±\sqrt{21^{2}-4\times 14}}{2\times 14}$$

Square 21.

$$m=\frac{-21±\sqrt{441-4\times 14}}{2\times 14}$$

Multiply -4 times 14.

$$m=\frac{-21±\sqrt{441-56}}{2\times 14}$$

Add 441 to -56.

$$m=\frac{-21±\sqrt{385}}{2\times 14}$$

Multiply 2 times 14.

$$m=\frac{-21±\sqrt{385}}{28}$$

Now solve the equation m=\frac{-21±\sqrt{385}}{28} when ± is plus. Add -21 to \sqrt{385}.

$$m=\frac{\sqrt{385}-21}{28}$$

Divide -21+\sqrt{385} by 28.

$$m=\frac{\sqrt{385}}{28}-\frac{3}{4}$$

Now solve the equation m=\frac{-21±\sqrt{385}}{28} when ± is minus. Subtract \sqrt{385} from -21.

$$m=\frac{-\sqrt{385}-21}{28}$$

Divide -21-\sqrt{385} by 28.

$$m=-\frac{\sqrt{385}}{28}-\frac{3}{4}$$

The equation is now solved.

$$m=\frac{\sqrt{385}}{28}-\frac{3}{4} m=-\frac{\sqrt{385}}{28}-\frac{3}{4}$$

Steps for Completing the Square

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

$$14m^{2}+21m+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$14m^{2}+21m=-1$$

Divide both sides by 14.

$$\frac{14m^{2}+21m}{14}=-\frac{1}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$m^{2}+\frac{21}{14}m=-\frac{1}{14}$$

Reduce the fraction \frac{21}{14} to lowest terms by extracting and canceling out 7.

$$m^{2}+\frac{3}{2}m=-\frac{1}{14}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{3}{2}m+\left(\frac{3}{4}\right)^{2}=-\frac{1}{14}+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=-\frac{1}{14}+\frac{9}{16}$$

Add -\frac{1}{14} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=\frac{55}{112}$$

Factor m^{2}+\frac{3}{2}m+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{3}{4}\right)^{2}=\frac{55}{112}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{3}{4}\right)^{2}}=\sqrt{\frac{55}{112}}$$

Simplify.

$$m+\frac{3}{4}=\frac{\sqrt{385}}{28} m+\frac{3}{4}=-\frac{\sqrt{385}}{28}$$

Subtract \frac{3}{4} from both sides of the equation.

$$m=\frac{\sqrt{385}}{28}-\frac{3}{4} m=-\frac{\sqrt{385}}{28}-\frac{3}{4}$$