Solve 12/11+m+12/15m=1/4

$$\frac { 12 } { 11 + m } + \frac { 12 } { 15 m } = \frac { 1 } { 4 }$$

Solve for m

$$m = \frac{\sqrt{43921} + 201}{10} \approx 41.057337617$$

$$m=\frac{201-\sqrt{43921}}{10}\approx -0.857337617$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by 60m\left(m+11\right), the least common multiple of 11+m,15m,4.

$$60m\times 12+\left(4m+44\right)\times 12=15m\left(m+11\right)$$

Multiply 60 and 12 to get 720.

$$720m+\left(4m+44\right)\times 12=15m\left(m+11\right)$$

Use the distributive property to multiply 4m+44 by 12.

$$720m+48m+528=15m\left(m+11\right)$$

Combine 720m and 48m to get 768m.

$$768m+528=15m\left(m+11\right)$$

Use the distributive property to multiply 15m by m+11.

$$768m+528=15m^{2}+165m$$

Subtract 15m^{2} from both sides.

$$768m+528-15m^{2}=165m$$

Subtract 165m from both sides.

$$768m+528-15m^{2}-165m=0$$

Combine 768m and -165m to get 603m.

$$603m+528-15m^{2}=0$$

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.

$$-15m^{2}+603m+528=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 603 for b, and 528 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-603±\sqrt{603^{2}-4\left(-15\right)\times 528}}{2\left(-15\right)}$$

Square 603.

$$m=\frac{-603±\sqrt{363609-4\left(-15\right)\times 528}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$m=\frac{-603±\sqrt{363609+60\times 528}}{2\left(-15\right)}$$

Multiply 60 times 528.

$$m=\frac{-603±\sqrt{363609+31680}}{2\left(-15\right)}$$

Add 363609 to 31680.

$$m=\frac{-603±\sqrt{395289}}{2\left(-15\right)}$$

Take the square root of 395289.

$$m=\frac{-603±3\sqrt{43921}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$m=\frac{-603±3\sqrt{43921}}{-30}$$

Now solve the equation m=\frac{-603±3\sqrt{43921}}{-30} when ± is plus. Add -603 to 3\sqrt{43921}.

$$m=\frac{3\sqrt{43921}-603}{-30}$$

Divide -603+3\sqrt{43921} by -30.

$$m=\frac{201-\sqrt{43921}}{10}$$

Now solve the equation m=\frac{-603±3\sqrt{43921}}{-30} when ± is minus. Subtract 3\sqrt{43921} from -603.

$$m=\frac{-3\sqrt{43921}-603}{-30}$$

Divide -603-3\sqrt{43921} by -30.

$$m=\frac{\sqrt{43921}+201}{10}$$

The equation is now solved.

$$m=\frac{201-\sqrt{43921}}{10} m=\frac{\sqrt{43921}+201}{10}$$

Steps for Completing the Square

Variable m cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by 60m\left(m+11\right), the least common multiple of 11+m,15m,4.

$$60m\times 12+\left(4m+44\right)\times 12=15m\left(m+11\right)$$

Multiply 60 and 12 to get 720.

$$720m+\left(4m+44\right)\times 12=15m\left(m+11\right)$$

Use the distributive property to multiply 4m+44 by 12.

$$720m+48m+528=15m\left(m+11\right)$$

Combine 720m and 48m to get 768m.

$$768m+528=15m\left(m+11\right)$$

Use the distributive property to multiply 15m by m+11.

$$768m+528=15m^{2}+165m$$

Subtract 15m^{2} from both sides.

$$768m+528-15m^{2}=165m$$

Subtract 165m from both sides.

$$768m+528-15m^{2}-165m=0$$

Combine 768m and -165m to get 603m.

$$603m+528-15m^{2}=0$$

Subtract 528 from both sides. Anything subtracted from zero gives its negation.

$$603m-15m^{2}=-528$$

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.

$$-15m^{2}+603m=-528$$

Divide both sides by -15.

$$\frac{-15m^{2}+603m}{-15}=-\frac{528}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$m^{2}+\frac{603}{-15}m=-\frac{528}{-15}$$

Reduce the fraction \frac{603}{-15} to lowest terms by extracting and canceling out 3.

$$m^{2}-\frac{201}{5}m=-\frac{528}{-15}$$

Reduce the fraction \frac{-528}{-15} to lowest terms by extracting and canceling out 3.

$$m^{2}-\frac{201}{5}m=\frac{176}{5}$$

Divide -\frac{201}{5}, the coefficient of the x term, by 2 to get -\frac{201}{10}. Then add the square of -\frac{201}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{201}{5}m+\left(-\frac{201}{10}\right)^{2}=\frac{176}{5}+\left(-\frac{201}{10}\right)^{2}$$

Square -\frac{201}{10} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{201}{5}m+\frac{40401}{100}=\frac{176}{5}+\frac{40401}{100}$$

Add \frac{176}{5} to \frac{40401}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{201}{5}m+\frac{40401}{100}=\frac{43921}{100}$$

Factor m^{2}-\frac{201}{5}m+\frac{40401}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{201}{10}\right)^{2}=\frac{43921}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{201}{10}\right)^{2}}=\sqrt{\frac{43921}{100}}$$

Simplify.

$$m-\frac{201}{10}=\frac{\sqrt{43921}}{10} m-\frac{201}{10}=-\frac{\sqrt{43921}}{10}$$

Add \frac{201}{10} to both sides of the equation.

$$m=\frac{\sqrt{43921}+201}{10} m=\frac{201-\sqrt{43921}}{10}$$