Solve 56/9-x-56/(9+x)=3

$$\frac{ 56 }{ 9-x } - \frac{ 56 }{ (9+x) } = 3$$

Solve for x

$$x = \frac{\sqrt{3865} - 56}{3} \approx 2.056374723$$

$$x=\frac{-\sqrt{3865}-56}{3}\approx -39.389708056$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x+9\right), the least common multiple of 9-x,9+x.

$$-\left(9+x\right)\times 56-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

To find the opposite of 9+x, find the opposite of each term.

$$\left(-9-x\right)\times 56-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply -9-x by 56.

$$-504-56x-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply x-9 by 56.

$$-504-56x-\left(56x-504\right)=3\left(x-9\right)\left(x+9\right)$$

To find the opposite of 56x-504, find the opposite of each term.

$$-504-56x-56x+504=3\left(x-9\right)\left(x+9\right)$$

Combine -56x and -56x to get -112x.

$$-504-112x+504=3\left(x-9\right)\left(x+9\right)$$

Add -504 and 504 to get 0.

$$-112x=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply 3 by x-9.

$$-112x=\left(3x-27\right)\left(x+9\right)$$

Use the distributive property to multiply 3x-27 by x+9 and combine like terms.

$$-112x=3x^{2}-243$$

Subtract 3x^{2} from both sides.

$$-112x-3x^{2}=-243$$

Add 243 to both sides.

$$-112x-3x^{2}+243=0$$

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.

$$-3x^{2}-112x+243=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -112 for b, and 243 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-3\right)\times 243}}{2\left(-3\right)}$$

Square -112.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\left(-3\right)\times 243}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-112\right)±\sqrt{12544+12\times 243}}{2\left(-3\right)}$$

Multiply 12 times 243.

$$x=\frac{-\left(-112\right)±\sqrt{12544+2916}}{2\left(-3\right)}$$

Add 12544 to 2916.

$$x=\frac{-\left(-112\right)±\sqrt{15460}}{2\left(-3\right)}$$

Take the square root of 15460.

$$x=\frac{-\left(-112\right)±2\sqrt{3865}}{2\left(-3\right)}$$

The opposite of -112 is 112.

$$x=\frac{112±2\sqrt{3865}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{112±2\sqrt{3865}}{-6}$$

Now solve the equation x=\frac{112±2\sqrt{3865}}{-6} when ± is plus. Add 112 to 2\sqrt{3865}.

$$x=\frac{2\sqrt{3865}+112}{-6}$$

Divide 112+2\sqrt{3865} by -6.

$$x=\frac{-\sqrt{3865}-56}{3}$$

Now solve the equation x=\frac{112±2\sqrt{3865}}{-6} when ± is minus. Subtract 2\sqrt{3865} from 112.

$$x=\frac{112-2\sqrt{3865}}{-6}$$

Divide 112-2\sqrt{3865} by -6.

$$x=\frac{\sqrt{3865}-56}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{3865}-56}{3} x=\frac{\sqrt{3865}-56}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x+9\right), the least common multiple of 9-x,9+x.

$$-\left(9+x\right)\times 56-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

To find the opposite of 9+x, find the opposite of each term.

$$\left(-9-x\right)\times 56-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply -9-x by 56.

$$-504-56x-\left(x-9\right)\times 56=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply x-9 by 56.

$$-504-56x-\left(56x-504\right)=3\left(x-9\right)\left(x+9\right)$$

To find the opposite of 56x-504, find the opposite of each term.

$$-504-56x-56x+504=3\left(x-9\right)\left(x+9\right)$$

Combine -56x and -56x to get -112x.

$$-504-112x+504=3\left(x-9\right)\left(x+9\right)$$

Add -504 and 504 to get 0.

$$-112x=3\left(x-9\right)\left(x+9\right)$$

Use the distributive property to multiply 3 by x-9.

$$-112x=\left(3x-27\right)\left(x+9\right)$$

Use the distributive property to multiply 3x-27 by x+9 and combine like terms.

$$-112x=3x^{2}-243$$

Subtract 3x^{2} from both sides.

$$-112x-3x^{2}=-243$$

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.

$$-3x^{2}-112x=-243$$

Divide both sides by -3.

$$\frac{-3x^{2}-112x}{-3}=-\frac{243}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{112}{-3}\right)x=-\frac{243}{-3}$$

Divide -112 by -3.

$$x^{2}+\frac{112}{3}x=-\frac{243}{-3}$$

Divide -243 by -3.

$$x^{2}+\frac{112}{3}x=81$$

Divide \frac{112}{3}, the coefficient of the x term, by 2 to get \frac{56}{3}. Then add the square of \frac{56}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{112}{3}x+\left(\frac{56}{3}\right)^{2}=81+\left(\frac{56}{3}\right)^{2}$$

Square \frac{56}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{112}{3}x+\frac{3136}{9}=81+\frac{3136}{9}$$

Add 81 to \frac{3136}{9}.

$$x^{2}+\frac{112}{3}x+\frac{3136}{9}=\frac{3865}{9}$$

Factor x^{2}+\frac{112}{3}x+\frac{3136}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{56}{3}\right)^{2}=\frac{3865}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{56}{3}\right)^{2}}=\sqrt{\frac{3865}{9}}$$

Simplify.

$$x+\frac{56}{3}=\frac{\sqrt{3865}}{3} x+\frac{56}{3}=-\frac{\sqrt{3865}}{3}$$

Subtract \frac{56}{3} from both sides of the equation.

$$x=\frac{\sqrt{3865}-56}{3} x=\frac{-\sqrt{3865}-56}{3}$$