Solve 4/4-m=1/6m+1+1

$$\frac { 4 } { 4 - m } = \frac { 1 } { 6 m + 1 } + 1$$

Solve for m

$$m=-1$$

$$m=\frac{2}{3}\approx 0.666666667$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values -\frac{1}{6},4 since division by zero is not defined. Multiply both sides of the equation by \left(m-4\right)\left(6m+1\right), the least common multiple of 4-m,6m+1.

$$\left(-1-6m\right)\times 4=m-4+\left(m-4\right)\left(6m+1\right)$$

Use the distributive property to multiply -1-6m by 4.

$$-4-24m=m-4+\left(m-4\right)\left(6m+1\right)$$

Use the distributive property to multiply m-4 by 6m+1 and combine like terms.

$$-4-24m=m-4+6m^{2}-23m-4$$

Combine m and -23m to get -22m.

$$-4-24m=-22m-4+6m^{2}-4$$

Subtract 4 from -4 to get -8.

$$-4-24m=-22m-8+6m^{2}$$

Add 22m to both sides.

$$-4-24m+22m=-8+6m^{2}$$

Combine -24m and 22m to get -2m.

$$-4-2m=-8+6m^{2}$$

Subtract -8 from both sides.

$$-4-2m-\left(-8\right)=6m^{2}$$

The opposite of -8 is 8.

$$-4-2m+8=6m^{2}$$

Subtract 6m^{2} from both sides.

$$-4-2m+8-6m^{2}=0$$

Add -4 and 8 to get 4.

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

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.

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

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

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

Square -2.

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

Multiply -4 times -6.

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

Multiply 24 times 4.

$$m=\frac{-\left(-2\right)±\sqrt{4+96}}{2\left(-6\right)}$$

Add 4 to 96.

$$m=\frac{-\left(-2\right)±\sqrt{100}}{2\left(-6\right)}$$

Take the square root of 100.

$$m=\frac{-\left(-2\right)±10}{2\left(-6\right)}$$

The opposite of -2 is 2.

$$m=\frac{2±10}{2\left(-6\right)}$$

Multiply 2 times -6.

$$m=\frac{2±10}{-12}$$

Now solve the equation m=\frac{2±10}{-12} when ± is plus. Add 2 to 10.

$$m=\frac{12}{-12}$$

Divide 12 by -12.

$$m=-1$$

Now solve the equation m=\frac{2±10}{-12} when ± is minus. Subtract 10 from 2.

$$m=-\frac{8}{-12}$$

Reduce the fraction \frac{-8}{-12} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

$$m=-1 m=\frac{2}{3}$$

Steps for Completing the Square

Variable m cannot be equal to any of the values -\frac{1}{6},4 since division by zero is not defined. Multiply both sides of the equation by \left(m-4\right)\left(6m+1\right), the least common multiple of 4-m,6m+1.

$$\left(-1-6m\right)\times 4=m-4+\left(m-4\right)\left(6m+1\right)$$

Use the distributive property to multiply -1-6m by 4.

$$-4-24m=m-4+\left(m-4\right)\left(6m+1\right)$$

Use the distributive property to multiply m-4 by 6m+1 and combine like terms.

$$-4-24m=m-4+6m^{2}-23m-4$$

Combine m and -23m to get -22m.

$$-4-24m=-22m-4+6m^{2}-4$$

Subtract 4 from -4 to get -8.

$$-4-24m=-22m-8+6m^{2}$$

Add 22m to both sides.

$$-4-24m+22m=-8+6m^{2}$$

Combine -24m and 22m to get -2m.

$$-4-2m=-8+6m^{2}$$

Subtract 6m^{2} from both sides.

$$-4-2m-6m^{2}=-8$$

Add 4 to both sides.

$$-2m-6m^{2}=-8+4$$

Add -8 and 4 to get -4.

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

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.

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

Divide both sides by -6.

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

Dividing by -6 undoes the multiplication by -6.

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

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

$$m^{2}+\frac{1}{3}m=-\frac{4}{-6}$$

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

$$m^{2}+\frac{1}{3}m=\frac{2}{3}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{1}{3}m+\left(\frac{1}{6}\right)^{2}=\frac{2}{3}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{1}{3}m+\frac{1}{36}=\frac{2}{3}+\frac{1}{36}$$

Add \frac{2}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{1}{3}m+\frac{1}{36}=\frac{25}{36}$$

Factor m^{2}+\frac{1}{3}m+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{1}{6}\right)^{2}=\frac{25}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{6}\right)^{2}}=\sqrt{\frac{25}{36}}$$

Simplify.

$$m+\frac{1}{6}=\frac{5}{6} m+\frac{1}{6}=-\frac{5}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$m=\frac{2}{3} m=-1$$