Solve 3m+1/3m-1/frac{-2{3m-1}}=m+3

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

Solve for m

$$m = -\frac{7}{5} = -1\frac{2}{5} = -1.4$$

Steps Using the Quadratic Formula

Variable m cannot be equal to \frac{1}{3} since division by zero is not defined. Divide \frac{3m+1}{3m-1} by \frac{-2}{3m-1} by multiplying \frac{3m+1}{3m-1} by the reciprocal of \frac{-2}{3m-1}.

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

Consider \left(3m+1\right)\left(3m-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Expand \left(3m\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{9m^{2}-1}{\left(3m-1\right)\left(-2\right)}=m+3$$

Use the distributive property to multiply 3m-1 by -2.

$$\frac{9m^{2}-1}{-6m+2}=m+3$$

Subtract m from both sides.

$$\frac{9m^{2}-1}{-6m+2}-m=3$$

Factor -6m+2.

$$\frac{9m^{2}-1}{2\left(-3m+1\right)}-m=3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply m times \frac{2\left(-3m+1\right)}{2\left(-3m+1\right)}.

$$\frac{9m^{2}-1}{2\left(-3m+1\right)}-\frac{m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=3$$

Since \frac{9m^{2}-1}{2\left(-3m+1\right)} and \frac{m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9m^{2}-1-m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=3$$

Do the multiplications in 9m^{2}-1-m\times 2\left(-3m+1\right).

$$\frac{9m^{2}-1+6m^{2}-2m}{2\left(-3m+1\right)}=3$$

Combine like terms in 9m^{2}-1+6m^{2}-2m.

$$\frac{15m^{2}-1-2m}{2\left(-3m+1\right)}=3$$

Subtract 3 from both sides.

$$\frac{15m^{2}-1-2m}{2\left(-3m+1\right)}-3=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2\left(-3m+1\right)}{2\left(-3m+1\right)}.

$$\frac{15m^{2}-1-2m}{2\left(-3m+1\right)}-\frac{3\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=0$$

Since \frac{15m^{2}-1-2m}{2\left(-3m+1\right)} and \frac{3\times 2\left(-3m+1\right)}{2\left(-3m+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15m^{2}-1-2m-3\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=0$$

Do the multiplications in 15m^{2}-1-2m-3\times 2\left(-3m+1\right).

$$\frac{15m^{2}-1-2m+18m-6}{2\left(-3m+1\right)}=0$$

Combine like terms in 15m^{2}-1-2m+18m-6.

$$\frac{15m^{2}-7+16m}{2\left(-3m+1\right)}=0$$

Variable m cannot be equal to \frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(-3m+1\right).

$$15m^{2}-7+16m=0$$

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.

$$15m^{2}+16m-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 16 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-16±\sqrt{16^{2}-4\times 15\left(-7\right)}}{2\times 15}$$

Square 16.

$$m=\frac{-16±\sqrt{256-4\times 15\left(-7\right)}}{2\times 15}$$

Multiply -4 times 15.

$$m=\frac{-16±\sqrt{256-60\left(-7\right)}}{2\times 15}$$

Multiply -60 times -7.

$$m=\frac{-16±\sqrt{256+420}}{2\times 15}$$

Add 256 to 420.

$$m=\frac{-16±\sqrt{676}}{2\times 15}$$

Take the square root of 676.

$$m=\frac{-16±26}{2\times 15}$$

Multiply 2 times 15.

$$m=\frac{-16±26}{30}$$

Now solve the equation m=\frac{-16±26}{30} when ± is plus. Add -16 to 26.

$$m=\frac{10}{30}$$

Reduce the fraction \frac{10}{30} to lowest terms by extracting and canceling out 10.

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

Now solve the equation m=\frac{-16±26}{30} when ± is minus. Subtract 26 from -16.

$$m=-\frac{42}{30}$$

Reduce the fraction \frac{-42}{30} to lowest terms by extracting and canceling out 6.

$$m=-\frac{7}{5}$$

The equation is now solved.

$$m=\frac{1}{3} m=-\frac{7}{5}$$

Variable m cannot be equal to \frac{1}{3}.

$$m=-\frac{7}{5}$$

Steps for Completing the Square

Variable m cannot be equal to \frac{1}{3} since division by zero is not defined. Divide \frac{3m+1}{3m-1} by \frac{-2}{3m-1} by multiplying \frac{3m+1}{3m-1} by the reciprocal of \frac{-2}{3m-1}.

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

Consider \left(3m+1\right)\left(3m-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Expand \left(3m\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{9m^{2}-1}{\left(3m-1\right)\left(-2\right)}=m+3$$

Use the distributive property to multiply 3m-1 by -2.

$$\frac{9m^{2}-1}{-6m+2}=m+3$$

Subtract m from both sides.

$$\frac{9m^{2}-1}{-6m+2}-m=3$$

Factor -6m+2.

$$\frac{9m^{2}-1}{2\left(-3m+1\right)}-m=3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply m times \frac{2\left(-3m+1\right)}{2\left(-3m+1\right)}.

$$\frac{9m^{2}-1}{2\left(-3m+1\right)}-\frac{m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=3$$

Since \frac{9m^{2}-1}{2\left(-3m+1\right)} and \frac{m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9m^{2}-1-m\times 2\left(-3m+1\right)}{2\left(-3m+1\right)}=3$$

Do the multiplications in 9m^{2}-1-m\times 2\left(-3m+1\right).

$$\frac{9m^{2}-1+6m^{2}-2m}{2\left(-3m+1\right)}=3$$

Combine like terms in 9m^{2}-1+6m^{2}-2m.

$$\frac{15m^{2}-1-2m}{2\left(-3m+1\right)}=3$$

Variable m cannot be equal to \frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(-3m+1\right).

$$15m^{2}-1-2m=6\left(-3m+1\right)$$

Use the distributive property to multiply 6 by -3m+1.

$$15m^{2}-1-2m=-18m+6$$

Add 18m to both sides.

$$15m^{2}-1-2m+18m=6$$

Combine -2m and 18m to get 16m.

$$15m^{2}-1+16m=6$$

Add 1 to both sides.

$$15m^{2}+16m=6+1$$

Add 6 and 1 to get 7.

$$15m^{2}+16m=7$$

Divide both sides by 15.

$$\frac{15m^{2}+16m}{15}=\frac{7}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$m^{2}+\frac{16}{15}m=\frac{7}{15}$$

Divide \frac{16}{15}, the coefficient of the x term, by 2 to get \frac{8}{15}. Then add the square of \frac{8}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{16}{15}m+\left(\frac{8}{15}\right)^{2}=\frac{7}{15}+\left(\frac{8}{15}\right)^{2}$$

Square \frac{8}{15} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{16}{15}m+\frac{64}{225}=\frac{7}{15}+\frac{64}{225}$$

Add \frac{7}{15} to \frac{64}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{16}{15}m+\frac{64}{225}=\frac{169}{225}$$

Factor m^{2}+\frac{16}{15}m+\frac{64}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{8}{15}\right)^{2}=\frac{169}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{8}{15}\right)^{2}}=\sqrt{\frac{169}{225}}$$

Simplify.

$$m+\frac{8}{15}=\frac{13}{15} m+\frac{8}{15}=-\frac{13}{15}$$

Subtract \frac{8}{15} from both sides of the equation.

$$m=\frac{1}{3} m=-\frac{7}{5}$$

Variable m cannot be equal to \frac{1}{3}.

$$m=-\frac{7}{5}$$