Solve 72/x+3+33/x-3=39

$$\frac { 72 } { x + 3 } + \frac { 33 } { x - 3 } = 39$$

Solve for x

$$x = \frac{\sqrt{5281} + 35}{26} \approx 4.141172661$$

$$x=\frac{35-\sqrt{5281}}{26}\approx -1.448864969$$

Steps Using the Quadratic Formula

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

$$\left(x-3\right)\times 72+\left(x+3\right)\times 33=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by 72.

$$72x-216+\left(x+3\right)\times 33=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 33.

$$72x-216+33x+99=39\left(x-3\right)\left(x+3\right)$$

Combine 72x and 33x to get 105x.

$$105x-216+99=39\left(x-3\right)\left(x+3\right)$$

Add -216 and 99 to get -117.

$$105x-117=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 39 by x-3.

$$105x-117=\left(39x-117\right)\left(x+3\right)$$

Use the distributive property to multiply 39x-117 by x+3 and combine like terms.

$$105x-117=39x^{2}-351$$

Subtract 39x^{2} from both sides.

$$105x-117-39x^{2}=-351$$

Add 351 to both sides.

$$105x-117-39x^{2}+351=0$$

Add -117 and 351 to get 234.

$$105x+234-39x^{2}=0$$

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.

$$-39x^{2}+105x+234=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -39 for a, 105 for b, and 234 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-105±\sqrt{105^{2}-4\left(-39\right)\times 234}}{2\left(-39\right)}$$

Square 105.

$$x=\frac{-105±\sqrt{11025-4\left(-39\right)\times 234}}{2\left(-39\right)}$$

Multiply -4 times -39.

$$x=\frac{-105±\sqrt{11025+156\times 234}}{2\left(-39\right)}$$

Multiply 156 times 234.

$$x=\frac{-105±\sqrt{11025+36504}}{2\left(-39\right)}$$

Add 11025 to 36504.

$$x=\frac{-105±\sqrt{47529}}{2\left(-39\right)}$$

Take the square root of 47529.

$$x=\frac{-105±3\sqrt{5281}}{2\left(-39\right)}$$

Multiply 2 times -39.

$$x=\frac{-105±3\sqrt{5281}}{-78}$$

Now solve the equation x=\frac{-105±3\sqrt{5281}}{-78} when ± is plus. Add -105 to 3\sqrt{5281}.

$$x=\frac{3\sqrt{5281}-105}{-78}$$

Divide -105+3\sqrt{5281} by -78.

$$x=\frac{35-\sqrt{5281}}{26}$$

Now solve the equation x=\frac{-105±3\sqrt{5281}}{-78} when ± is minus. Subtract 3\sqrt{5281} from -105.

$$x=\frac{-3\sqrt{5281}-105}{-78}$$

Divide -105-3\sqrt{5281} by -78.

$$x=\frac{\sqrt{5281}+35}{26}$$

The equation is now solved.

$$x=\frac{35-\sqrt{5281}}{26} x=\frac{\sqrt{5281}+35}{26}$$

Steps for Completing the Square

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

$$\left(x-3\right)\times 72+\left(x+3\right)\times 33=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by 72.

$$72x-216+\left(x+3\right)\times 33=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 33.

$$72x-216+33x+99=39\left(x-3\right)\left(x+3\right)$$

Combine 72x and 33x to get 105x.

$$105x-216+99=39\left(x-3\right)\left(x+3\right)$$

Add -216 and 99 to get -117.

$$105x-117=39\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 39 by x-3.

$$105x-117=\left(39x-117\right)\left(x+3\right)$$

Use the distributive property to multiply 39x-117 by x+3 and combine like terms.

$$105x-117=39x^{2}-351$$

Subtract 39x^{2} from both sides.

$$105x-117-39x^{2}=-351$$

Add 117 to both sides.

$$105x-39x^{2}=-351+117$$

Add -351 and 117 to get -234.

$$105x-39x^{2}=-234$$

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.

$$-39x^{2}+105x=-234$$

Divide both sides by -39.

$$\frac{-39x^{2}+105x}{-39}=-\frac{234}{-39}$$

Dividing by -39 undoes the multiplication by -39.

$$x^{2}+\frac{105}{-39}x=-\frac{234}{-39}$$

Reduce the fraction \frac{105}{-39} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{35}{13}x=-\frac{234}{-39}$$

Divide -234 by -39.

$$x^{2}-\frac{35}{13}x=6$$

Divide -\frac{35}{13}, the coefficient of the x term, by 2 to get -\frac{35}{26}. Then add the square of -\frac{35}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{13}x+\left(-\frac{35}{26}\right)^{2}=6+\left(-\frac{35}{26}\right)^{2}$$

Square -\frac{35}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{13}x+\frac{1225}{676}=6+\frac{1225}{676}$$

Add 6 to \frac{1225}{676}.

$$x^{2}-\frac{35}{13}x+\frac{1225}{676}=\frac{5281}{676}$$

Factor x^{2}-\frac{35}{13}x+\frac{1225}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{26}\right)^{2}=\frac{5281}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{26}\right)^{2}}=\sqrt{\frac{5281}{676}}$$

Simplify.

$$x-\frac{35}{26}=\frac{\sqrt{5281}}{26} x-\frac{35}{26}=-\frac{\sqrt{5281}}{26}$$

Add \frac{35}{26} to both sides of the equation.

$$x=\frac{\sqrt{5281}+35}{26} x=\frac{35-\sqrt{5281}}{26}$$