Solve 2m^2+16/5=2m

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

Solve for m

$$m=\frac{5+\sqrt{7}i}{2}\approx 2.5+1.322875656i$$

$$m=\frac{-\sqrt{7}i+5}{2}\approx 2.5-1.322875656i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$2m^{2}+16=10m$$

Subtract 10m from both sides.

$$2m^{2}+16-10m=0$$

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.

$$2m^{2}-10m+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -10 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 2\times 16}}{2\times 2}$$

Square -10.

$$m=\frac{-\left(-10\right)±\sqrt{100-4\times 2\times 16}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-10\right)±\sqrt{100-8\times 16}}{2\times 2}$$

Multiply -8 times 16.

$$m=\frac{-\left(-10\right)±\sqrt{100-128}}{2\times 2}$$

Add 100 to -128.

$$m=\frac{-\left(-10\right)±\sqrt{-28}}{2\times 2}$$

Take the square root of -28.

$$m=\frac{-\left(-10\right)±2\sqrt{7}i}{2\times 2}$$

The opposite of -10 is 10.

$$m=\frac{10±2\sqrt{7}i}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{10±2\sqrt{7}i}{4}$$

Now solve the equation m=\frac{10±2\sqrt{7}i}{4} when ± is plus. Add 10 to 2i\sqrt{7}.

$$m=\frac{10+2\sqrt{7}i}{4}$$

Divide 10+2i\sqrt{7} by 4.

$$m=\frac{5+\sqrt{7}i}{2}$$

Now solve the equation m=\frac{10±2\sqrt{7}i}{4} when ± is minus. Subtract 2i\sqrt{7} from 10.

$$m=\frac{-2\sqrt{7}i+10}{4}$$

Divide 10-2i\sqrt{7} by 4.

$$m=\frac{-\sqrt{7}i+5}{2}$$

The equation is now solved.

$$m=\frac{5+\sqrt{7}i}{2} m=\frac{-\sqrt{7}i+5}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$2m^{2}+16=10m$$

Subtract 10m from both sides.

$$2m^{2}+16-10m=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$2m^{2}-10m=-16$$

Divide both sides by 2.

$$\frac{2m^{2}-10m}{2}=-\frac{16}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\left(-\frac{10}{2}\right)m=-\frac{16}{2}$$

Divide -10 by 2.

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

Divide -16 by 2.

$$m^{2}-5m=-8$$

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

$$m^{2}-5m+\left(-\frac{5}{2}\right)^{2}=-8+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-5m+\frac{25}{4}=-8+\frac{25}{4}$$

Add -8 to \frac{25}{4}.

$$m^{2}-5m+\frac{25}{4}=-\frac{7}{4}$$

Factor m^{2}-5m+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m-\frac{5}{2}=\frac{\sqrt{7}i}{2} m-\frac{5}{2}=-\frac{\sqrt{7}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$m=\frac{5+\sqrt{7}i}{2} m=\frac{-\sqrt{7}i+5}{2}$$