Solve 2m^2=14+5m

$$2 m ^ { 2 } = 14 + 5 m$$

Solve for m

$$m = \frac{\sqrt{137} + 5}{4} \approx 4.176174978$$

$$m=\frac{5-\sqrt{137}}{4}\approx -1.676174978$$

Steps Using the Quadratic Formula

Subtract 14 from both sides.

$$2m^{2}-14=5m$$

Subtract 5m from both sides.

$$2m^{2}-14-5m=0$$

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.

$$2m^{2}-5m-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-14\right)}}{2\times 2}$$

Square -5.

$$m=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-14\right)}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-5\right)±\sqrt{25-8\left(-14\right)}}{2\times 2}$$

Multiply -8 times -14.

$$m=\frac{-\left(-5\right)±\sqrt{25+112}}{2\times 2}$$

Add 25 to 112.

$$m=\frac{-\left(-5\right)±\sqrt{137}}{2\times 2}$$

The opposite of -5 is 5.

$$m=\frac{5±\sqrt{137}}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{5±\sqrt{137}}{4}$$

Now solve the equation m=\frac{5±\sqrt{137}}{4} when ± is plus. Add 5 to \sqrt{137}.

$$m=\frac{\sqrt{137}+5}{4}$$

Now solve the equation m=\frac{5±\sqrt{137}}{4} when ± is minus. Subtract \sqrt{137} from 5.

$$m=\frac{5-\sqrt{137}}{4}$$

The equation is now solved.

$$m=\frac{\sqrt{137}+5}{4} m=\frac{5-\sqrt{137}}{4}$$

Steps for Completing the Square

Subtract 5m from both sides.

$$2m^{2}-5m=14$$

Divide both sides by 2.

$$\frac{2m^{2}-5m}{2}=\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}-\frac{5}{2}m=\frac{14}{2}$$

Divide 14 by 2.

$$m^{2}-\frac{5}{2}m=7$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{2}m+\left(-\frac{5}{4}\right)^{2}=7+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=7+\frac{25}{16}$$

Add 7 to \frac{25}{16}.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=\frac{137}{16}$$

Factor m^{2}-\frac{5}{2}m+\frac{25}{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}.

$$\left(m-\frac{5}{4}\right)^{2}=\frac{137}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{4}\right)^{2}}=\sqrt{\frac{137}{16}}$$

Simplify.

$$m-\frac{5}{4}=\frac{\sqrt{137}}{4} m-\frac{5}{4}=-\frac{\sqrt{137}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$m=\frac{\sqrt{137}+5}{4} m=\frac{5-\sqrt{137}}{4}$$