Solve em(6m-5)+4=00

$$e m ( 6 m - 5 ) + 4 = 00$$

Solve for m

$$m=\frac{5+i\sqrt{\frac{96}{e}-25}}{12}\approx 0.416666667+0.267659952i$$

$$m=\frac{-i\sqrt{\frac{96}{e}-25}+5}{12}\approx 0.416666667-0.267659952i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply em by 6m-5.

$$6em^{2}-5em+4=0\times 0$$

Multiply 0 and 0 to get 0.

$$6em^{2}-5em+4=0$$

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.

$$6em^{2}+\left(-5e\right)m+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6e for a, -5e for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-5e\right)±\sqrt{\left(-5e\right)^{2}-4\times 6e\times 4}}{2\times 6e}$$

Square -5e.

$$m=\frac{-\left(-5e\right)±\sqrt{25e^{2}-4\times 6e\times 4}}{2\times 6e}$$

Multiply -4 times 6e.

$$m=\frac{-\left(-5e\right)±\sqrt{25e^{2}+\left(-24e\right)\times 4}}{2\times 6e}$$

Multiply -24e times 4.

$$m=\frac{-\left(-5e\right)±\sqrt{25e^{2}-96e}}{2\times 6e}$$

Add 25e^{2} to -96e.

$$m=\frac{-\left(-5e\right)±\sqrt{e\left(25e-96\right)}}{2\times 6e}$$

Take the square root of e\left(25e-96\right).

$$m=\frac{-\left(-5e\right)±i\sqrt{e\left(96-25e\right)}}{2\times 6e}$$

The opposite of -5e is 5e.

$$m=\frac{5e±i\sqrt{e\left(96-25e\right)}}{2\times 6e}$$

Multiply 2 times 6e.

$$m=\frac{5e±i\sqrt{e\left(96-25e\right)}}{12e}$$

Now solve the equation m=\frac{5e±i\sqrt{e\left(96-25e\right)}}{12e} when ± is plus. Add 5e to i\sqrt{e\left(-25e+96\right)}.

$$m=\frac{5e+i\sqrt{e\left(96-25e\right)}}{12e}$$

Divide 5e+i\sqrt{e\left(-25e+96\right)} by 12e.

$$m=\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12}$$

Now solve the equation m=\frac{5e±i\sqrt{e\left(96-25e\right)}}{12e} when ± is minus. Subtract i\sqrt{e\left(-25e+96\right)} from 5e.

$$m=\frac{-i\sqrt{e\left(96-25e\right)}+5e}{12e}$$

Divide 5e-i\sqrt{e\left(-25e+96\right)} by 12e.

$$m=-\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12}$$

The equation is now solved.

$$m=\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12} m=-\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12}$$

Steps for Completing the Square

Use the distributive property to multiply em by 6m-5.

$$6em^{2}-5em+4=0\times 0$$

Multiply 0 and 0 to get 0.

$$6em^{2}-5em+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$6em^{2}-5em=-4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6em^{2}+\left(-5e\right)m=-4$$

Divide both sides by 6e.

$$\frac{6em^{2}+\left(-5e\right)m}{6e}=-\frac{4}{6e}$$

Dividing by 6e undoes the multiplication by 6e.

$$m^{2}+\left(-\frac{5e}{6e}\right)m=-\frac{4}{6e}$$

Divide -5e by 6e.

$$m^{2}-\frac{5}{6}m=-\frac{4}{6e}$$

Divide -4 by 6e.

$$m^{2}-\frac{5}{6}m=-\frac{2}{3e}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{6}m+\left(-\frac{5}{12}\right)^{2}=-\frac{2}{3e}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{6}m+\frac{25}{144}=-\frac{2}{3e}+\frac{25}{144}$$

Factor m^{2}-\frac{5}{6}m+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{5}{12}\right)^{2}=-\frac{2}{3e}+\frac{25}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{12}\right)^{2}}=\sqrt{-\frac{2}{3e}+\frac{25}{144}}$$

Simplify.

$$m-\frac{5}{12}=\frac{i\sqrt{96-25e}}{12\sqrt{e}} m-\frac{5}{12}=-\frac{i\sqrt{96-25e}}{12\sqrt{e}}$$

Add \frac{5}{12} to both sides of the equation.

$$m=\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12} m=-\frac{i\sqrt{96-25e}}{12\sqrt{e}}+\frac{5}{12}$$