Solve 1/m-4=m/m+24

$$\frac { 1 } { m - 4 } = \frac { m } { m + 24 }$$

Solve for m

$$m=-3$$

$$m=8$$

Steps Using Factoring By Grouping

Variable m cannot be equal to any of the values -24,4 since division by zero is not defined. Multiply both sides of the equation by \left(m-4\right)\left(m+24\right), the least common multiple of m-4,m+24.

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

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

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

Subtract m^{2} from both sides.

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

Add 4m to both sides.

$$m+24-m^{2}+4m=0$$

Combine m and 4m to get 5m.

$$5m+24-m^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-m^{2}+5m+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -m^{2}+am+bm+24. To find a and b, set up a system to be solved.

$$a+b=5 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=8 b=-3$$

Rewrite -m^{2}+5m+24 as \left(-m^{2}+8m\right)+\left(-3m+24\right).

$$\left(-m^{2}+8m\right)+\left(-3m+24\right)$$

Factor out -m in the first and -3 in the second group.

$$-m\left(m-8\right)-3\left(m-8\right)$$

Factor out common term m-8 by using distributive property.

$$\left(m-8\right)\left(-m-3\right)$$

To find equation solutions, solve m-8=0 and -m-3=0.

$$m=8 m=-3$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values -24,4 since division by zero is not defined. Multiply both sides of the equation by \left(m-4\right)\left(m+24\right), the least common multiple of m-4,m+24.

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

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

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

Subtract m^{2} from both sides.

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

Add 4m to both sides.

$$m+24-m^{2}+4m=0$$

Combine m and 4m to get 5m.

$$5m+24-m^{2}=0$$

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.

$$-m^{2}+5m+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 5.

$$m=\frac{-5±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-5±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$m=\frac{-5±\sqrt{25+96}}{2\left(-1\right)}$$

Add 25 to 96.

$$m=\frac{-5±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$m=\frac{-5±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-5±11}{-2}$$

Now solve the equation m=\frac{-5±11}{-2} when ± is plus. Add -5 to 11.

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

Divide 6 by -2.

$$m=-3$$

Now solve the equation m=\frac{-5±11}{-2} when ± is minus. Subtract 11 from -5.

$$m=-\frac{16}{-2}$$

Divide -16 by -2.

$$m=8$$

The equation is now solved.

$$m=-3 m=8$$

Steps for Completing the Square

Variable m cannot be equal to any of the values -24,4 since division by zero is not defined. Multiply both sides of the equation by \left(m-4\right)\left(m+24\right), the least common multiple of m-4,m+24.

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

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

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

Subtract m^{2} from both sides.

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

Add 4m to both sides.

$$m+24-m^{2}+4m=0$$

Combine m and 4m to get 5m.

$$5m+24-m^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

$$\frac{-m^{2}+5m}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\frac{5}{-1}m=-\frac{24}{-1}$$

Divide 5 by -1.

$$m^{2}-5m=-\frac{24}{-1}$$

Divide -24 by -1.

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

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-5m+\left(-\frac{5}{2}\right)^{2}=24+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-5m+\frac{25}{4}=24+\frac{25}{4}$$

Add 24 to \frac{25}{4}.

$$m^{2}-5m+\frac{25}{4}=\frac{121}{4}$$

Factor m^{2}-5m+\frac{25}{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}.

$$\left(m-\frac{5}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$m-\frac{5}{2}=\frac{11}{2} m-\frac{5}{2}=-\frac{11}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$m=8 m=-3$$