Solve 2m^2+3=6m

$$2 m ^ { 2 } + 3 = 6 m$$

Solve for m

$$m = \frac{\sqrt{3} + 3}{2} \approx 2.366025404$$

$$m=\frac{3-\sqrt{3}}{2}\approx 0.633974596$$

Steps Using the Quadratic Formula

Subtract 6m from both sides.

$$2m^{2}+3-6m=0$$

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.

$$2m^{2}-6m+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -6 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Square -6.

$$m=\frac{-\left(-6\right)±\sqrt{36-4\times 2\times 3}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-6\right)±\sqrt{36-8\times 3}}{2\times 2}$$

Multiply -8 times 3.

$$m=\frac{-\left(-6\right)±\sqrt{36-24}}{2\times 2}$$

Add 36 to -24.

$$m=\frac{-\left(-6\right)±\sqrt{12}}{2\times 2}$$

Take the square root of 12.

$$m=\frac{-\left(-6\right)±2\sqrt{3}}{2\times 2}$$

The opposite of -6 is 6.

$$m=\frac{6±2\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{6±2\sqrt{3}}{4}$$

Now solve the equation m=\frac{6±2\sqrt{3}}{4} when ± is plus. Add 6 to 2\sqrt{3}.

$$m=\frac{2\sqrt{3}+6}{4}$$

Divide 6+2\sqrt{3} by 4.

$$m=\frac{\sqrt{3}+3}{2}$$

Now solve the equation m=\frac{6±2\sqrt{3}}{4} when ± is minus. Subtract 2\sqrt{3} from 6.

$$m=\frac{6-2\sqrt{3}}{4}$$

Divide 6-2\sqrt{3} by 4.

$$m=\frac{3-\sqrt{3}}{2}$$

The equation is now solved.

$$m=\frac{\sqrt{3}+3}{2} m=\frac{3-\sqrt{3}}{2}$$

Steps for Completing the Square

Subtract 6m from both sides.

$$2m^{2}+3-6m=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$2m^{2}-6m=-3$$

Divide both sides by 2.

$$\frac{2m^{2}-6m}{2}=-\frac{3}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\left(-\frac{6}{2}\right)m=-\frac{3}{2}$$

Divide -6 by 2.

$$m^{2}-3m=-\frac{3}{2}$$

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-3m+\frac{9}{4}=-\frac{3}{2}+\frac{9}{4}$$

Add -\frac{3}{2} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-3m+\frac{9}{4}=\frac{3}{4}$$

Factor m^{2}-3m+\frac{9}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m-\frac{3}{2}=\frac{\sqrt{3}}{2} m-\frac{3}{2}=-\frac{\sqrt{3}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$m=\frac{\sqrt{3}+3}{2} m=\frac{3-\sqrt{3}}{2}$$