Solve 12/m-2-6/m-3=1

$$\frac { 12 } { m - 2 } - \frac { 6 } { m - 3 } = 1$$

Solve for m

$$m=5$$

$$m=6$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(m-3\right)\left(m-2\right), the least common multiple of m-2,m-3.

$$\left(m-3\right)\times 12-\left(m-2\right)\times 6=\left(m-3\right)\left(m-2\right)$$

Use the distributive property to multiply m-3 by 12.

$$12m-36-\left(m-2\right)\times 6=\left(m-3\right)\left(m-2\right)$$

Use the distributive property to multiply m-2 by 6.

$$12m-36-\left(6m-12\right)=\left(m-3\right)\left(m-2\right)$$

To find the opposite of 6m-12, find the opposite of each term.

$$12m-36-6m+12=\left(m-3\right)\left(m-2\right)$$

Combine 12m and -6m to get 6m.

$$6m-36+12=\left(m-3\right)\left(m-2\right)$$

Add -36 and 12 to get -24.

$$6m-24=\left(m-3\right)\left(m-2\right)$$

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

$$6m-24=m^{2}-5m+6$$

Subtract m^{2} from both sides.

$$6m-24-m^{2}=-5m+6$$

Add 5m to both sides.

$$6m-24-m^{2}+5m=6$$

Combine 6m and 5m to get 11m.

$$11m-24-m^{2}=6$$

Subtract 6 from both sides.

$$11m-24-m^{2}-6=0$$

Subtract 6 from -24 to get -30.

$$11m-30-m^{2}=0$$

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.

$$-m^{2}+11m-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\left(-30\right)}}{2\left(-1\right)}$$

Square 11.

$$m=\frac{-11±\sqrt{121-4\left(-1\right)\left(-30\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-11±\sqrt{121+4\left(-30\right)}}{2\left(-1\right)}$$

Multiply 4 times -30.

$$m=\frac{-11±\sqrt{121-120}}{2\left(-1\right)}$$

Add 121 to -120.

$$m=\frac{-11±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$m=\frac{-11±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-11±1}{-2}$$

Now solve the equation m=\frac{-11±1}{-2} when ± is plus. Add -11 to 1.

$$m=-\frac{10}{-2}$$

Divide -10 by -2.

$$m=5$$

Now solve the equation m=\frac{-11±1}{-2} when ± is minus. Subtract 1 from -11.

$$m=-\frac{12}{-2}$$

Divide -12 by -2.

$$m=6$$

The equation is now solved.

$$m=5 m=6$$

Steps for Completing the Square

Variable m cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(m-3\right)\left(m-2\right), the least common multiple of m-2,m-3.

$$\left(m-3\right)\times 12-\left(m-2\right)\times 6=\left(m-3\right)\left(m-2\right)$$

Use the distributive property to multiply m-3 by 12.

$$12m-36-\left(m-2\right)\times 6=\left(m-3\right)\left(m-2\right)$$

Use the distributive property to multiply m-2 by 6.

$$12m-36-\left(6m-12\right)=\left(m-3\right)\left(m-2\right)$$

To find the opposite of 6m-12, find the opposite of each term.

$$12m-36-6m+12=\left(m-3\right)\left(m-2\right)$$

Combine 12m and -6m to get 6m.

$$6m-36+12=\left(m-3\right)\left(m-2\right)$$

Add -36 and 12 to get -24.

$$6m-24=\left(m-3\right)\left(m-2\right)$$

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

$$6m-24=m^{2}-5m+6$$

Subtract m^{2} from both sides.

$$6m-24-m^{2}=-5m+6$$

Add 5m to both sides.

$$6m-24-m^{2}+5m=6$$

Combine 6m and 5m to get 11m.

$$11m-24-m^{2}=6$$

Add 24 to both sides.

$$11m-m^{2}=6+24$$

Add 6 and 24 to get 30.

$$11m-m^{2}=30$$

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.

$$-m^{2}+11m=30$$

Divide both sides by -1.

$$\frac{-m^{2}+11m}{-1}=\frac{30}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\frac{11}{-1}m=\frac{30}{-1}$$

Divide 11 by -1.

$$m^{2}-11m=\frac{30}{-1}$$

Divide 30 by -1.

$$m^{2}-11m=-30$$

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

$$m^{2}-11m+\left(-\frac{11}{2}\right)^{2}=-30+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-11m+\frac{121}{4}=-30+\frac{121}{4}$$

Add -30 to \frac{121}{4}.

$$m^{2}-11m+\frac{121}{4}=\frac{1}{4}$$

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

$$\left(m-\frac{11}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{11}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$m-\frac{11}{2}=\frac{1}{2} m-\frac{11}{2}=-\frac{1}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$m=6 m=5$$