Solve 2m-5/3m=2m/m-3

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

Solve for m

$$m = -\frac{15}{4} = -3\frac{3}{4} = -3.75$$

$$m=1$$

Steps Using Factoring By Grouping

Variable m cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by 3m\left(m-3\right), the least common multiple of 3m,m-3.

$$\left(m-3\right)\left(2m-5\right)=3m\times 2m$$

Use the distributive property to multiply m-3 by 2m-5 and combine like terms.

$$2m^{2}-11m+15=3m\times 2m$$

Multiply m and m to get m^{2}.

$$2m^{2}-11m+15=3m^{2}\times 2$$

Multiply 3 and 2 to get 6.

$$2m^{2}-11m+15=6m^{2}$$

Subtract 6m^{2} from both sides.

$$2m^{2}-11m+15-6m^{2}=0$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-11m+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4m^{2}+am+bm+15. To find a and b, set up a system to be solved.

$$a+b=-11 ab=-4\times 15=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -11.

$$a=4 b=-15$$

Rewrite -4m^{2}-11m+15 as \left(-4m^{2}+4m\right)+\left(-15m+15\right).

$$\left(-4m^{2}+4m\right)+\left(-15m+15\right)$$

Factor out 4m in the first and 15 in the second group.

$$4m\left(-m+1\right)+15\left(-m+1\right)$$

Factor out common term -m+1 by using distributive property.

$$\left(-m+1\right)\left(4m+15\right)$$

To find equation solutions, solve -m+1=0 and 4m+15=0.

$$m=1 m=-\frac{15}{4}$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by 3m\left(m-3\right), the least common multiple of 3m,m-3.

$$\left(m-3\right)\left(2m-5\right)=3m\times 2m$$

Use the distributive property to multiply m-3 by 2m-5 and combine like terms.

$$2m^{2}-11m+15=3m\times 2m$$

Multiply m and m to get m^{2}.

$$2m^{2}-11m+15=3m^{2}\times 2$$

Multiply 3 and 2 to get 6.

$$2m^{2}-11m+15=6m^{2}$$

Subtract 6m^{2} from both sides.

$$2m^{2}-11m+15-6m^{2}=0$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-11m+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -11 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-4\right)\times 15}}{2\left(-4\right)}$$

Square -11.

$$m=\frac{-\left(-11\right)±\sqrt{121-4\left(-4\right)\times 15}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$m=\frac{-\left(-11\right)±\sqrt{121+16\times 15}}{2\left(-4\right)}$$

Multiply 16 times 15.

$$m=\frac{-\left(-11\right)±\sqrt{121+240}}{2\left(-4\right)}$$

Add 121 to 240.

$$m=\frac{-\left(-11\right)±\sqrt{361}}{2\left(-4\right)}$$

Take the square root of 361.

$$m=\frac{-\left(-11\right)±19}{2\left(-4\right)}$$

The opposite of -11 is 11.

$$m=\frac{11±19}{2\left(-4\right)}$$

Multiply 2 times -4.

$$m=\frac{11±19}{-8}$$

Now solve the equation m=\frac{11±19}{-8} when ± is plus. Add 11 to 19.

$$m=\frac{30}{-8}$$

Reduce the fraction \frac{30}{-8} to lowest terms by extracting and canceling out 2.

$$m=-\frac{15}{4}$$

Now solve the equation m=\frac{11±19}{-8} when ± is minus. Subtract 19 from 11.

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

Divide -8 by -8.

$$m=1$$

The equation is now solved.

$$m=-\frac{15}{4} m=1$$

Steps for Completing the Square

Variable m cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by 3m\left(m-3\right), the least common multiple of 3m,m-3.

$$\left(m-3\right)\left(2m-5\right)=3m\times 2m$$

Use the distributive property to multiply m-3 by 2m-5 and combine like terms.

$$2m^{2}-11m+15=3m\times 2m$$

Multiply m and m to get m^{2}.

$$2m^{2}-11m+15=3m^{2}\times 2$$

Multiply 3 and 2 to get 6.

$$2m^{2}-11m+15=6m^{2}$$

Subtract 6m^{2} from both sides.

$$2m^{2}-11m+15-6m^{2}=0$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-11m+15=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$-4m^{2}-11m=-15$$

Divide both sides by -4.

$$\frac{-4m^{2}-11m}{-4}=-\frac{15}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$m^{2}+\left(-\frac{11}{-4}\right)m=-\frac{15}{-4}$$

Divide -11 by -4.

$$m^{2}+\frac{11}{4}m=-\frac{15}{-4}$$

Divide -15 by -4.

$$m^{2}+\frac{11}{4}m=\frac{15}{4}$$

Divide \frac{11}{4}, the coefficient of the x term, by 2 to get \frac{11}{8}. Then add the square of \frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{11}{4}m+\left(\frac{11}{8}\right)^{2}=\frac{15}{4}+\left(\frac{11}{8}\right)^{2}$$

Square \frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{11}{4}m+\frac{121}{64}=\frac{15}{4}+\frac{121}{64}$$

Add \frac{15}{4} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{11}{4}m+\frac{121}{64}=\frac{361}{64}$$

Factor m^{2}+\frac{11}{4}m+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{11}{8}\right)^{2}=\frac{361}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{11}{8}\right)^{2}}=\sqrt{\frac{361}{64}}$$

Simplify.

$$m+\frac{11}{8}=\frac{19}{8} m+\frac{11}{8}=-\frac{19}{8}$$

Subtract \frac{11}{8} from both sides of the equation.

$$m=1 m=-\frac{15}{4}$$