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

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

Solve for m

$$m=\frac{5+\sqrt{71}i}{12}\approx 0.416666667+0.702179148i$$

$$m=\frac{-\sqrt{71}i+5}{12}\approx 0.416666667-0.702179148i$$

Steps Using the Quadratic Formula

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

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

Multiply 0 and 0 to get 0.

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

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

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

Square -5.

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

Multiply -4 times 6.

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

Multiply -24 times 4.

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

Add 25 to -96.

$$m=\frac{-\left(-5\right)±\sqrt{-71}}{2\times 6}$$

Take the square root of -71.

$$m=\frac{-\left(-5\right)±\sqrt{71}i}{2\times 6}$$

The opposite of -5 is 5.

$$m=\frac{5±\sqrt{71}i}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{5±\sqrt{71}i}{12}$$

Now solve the equation m=\frac{5±\sqrt{71}i}{12} when ± is plus. Add 5 to i\sqrt{71}.

$$m=\frac{5+\sqrt{71}i}{12}$$

Now solve the equation m=\frac{5±\sqrt{71}i}{12} when ± is minus. Subtract i\sqrt{71} from 5.

$$m=\frac{-\sqrt{71}i+5}{12}$$

The equation is now solved.

$$m=\frac{5+\sqrt{71}i}{12} m=\frac{-\sqrt{71}i+5}{12}$$

Steps for Completing the Square

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

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

Multiply 0 and 0 to get 0.

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

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

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

Divide both sides by 6.

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

Dividing by 6 undoes the multiplication by 6.

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

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

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

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.

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

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

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

Add -\frac{2}{3} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{5}{6}m+\frac{25}{144}=-\frac{71}{144}$$

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}.

$$\left(m-\frac{5}{12}\right)^{2}=-\frac{71}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{12}\right)^{2}}=\sqrt{-\frac{71}{144}}$$

Simplify.

$$m-\frac{5}{12}=\frac{\sqrt{71}i}{12} m-\frac{5}{12}=-\frac{\sqrt{71}i}{12}$$

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

$$m=\frac{5+\sqrt{71}i}{12} m=\frac{-\sqrt{71}i+5}{12}$$