Solve 77/x+4+77/x-4=9

$$\frac{ 77 }{ x+4 } + \frac{ 77 }{ x-4 } =9$$

Solve for x

$$x=18$$

$$x=-\frac{8}{9}\approx -0.888888889$$

Steps Using the Quadratic Formula

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

$$\left(x-4\right)\times 77+\left(x+4\right)\times 77=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 77.

$$77x-308+\left(x+4\right)\times 77=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 77.

$$77x-308+77x+308=9\left(x-4\right)\left(x+4\right)$$

Combine 77x and 77x to get 154x.

$$154x-308+308=9\left(x-4\right)\left(x+4\right)$$

Add -308 and 308 to get 0.

$$154x=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 9 by x-4.

$$154x=\left(9x-36\right)\left(x+4\right)$$

Use the distributive property to multiply 9x-36 by x+4 and combine like terms.

$$154x=9x^{2}-144$$

Subtract 9x^{2} from both sides.

$$154x-9x^{2}=-144$$

Add 144 to both sides.

$$154x-9x^{2}+144=0$$

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.

$$-9x^{2}+154x+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 154 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-154±\sqrt{154^{2}-4\left(-9\right)\times 144}}{2\left(-9\right)}$$

Square 154.

$$x=\frac{-154±\sqrt{23716-4\left(-9\right)\times 144}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-154±\sqrt{23716+36\times 144}}{2\left(-9\right)}$$

Multiply 36 times 144.

$$x=\frac{-154±\sqrt{23716+5184}}{2\left(-9\right)}$$

Add 23716 to 5184.

$$x=\frac{-154±\sqrt{28900}}{2\left(-9\right)}$$

Take the square root of 28900.

$$x=\frac{-154±170}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-154±170}{-18}$$

Now solve the equation x=\frac{-154±170}{-18} when ± is plus. Add -154 to 170.

$$x=\frac{16}{-18}$$

Reduce the fraction \frac{16}{-18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{8}{9}$$

Now solve the equation x=\frac{-154±170}{-18} when ± is minus. Subtract 170 from -154.

$$x=-\frac{324}{-18}$$

Divide -324 by -18.

$$x=18$$

The equation is now solved.

$$x=-\frac{8}{9} x=18$$

Steps for Completing the Square

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

$$\left(x-4\right)\times 77+\left(x+4\right)\times 77=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 77.

$$77x-308+\left(x+4\right)\times 77=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 77.

$$77x-308+77x+308=9\left(x-4\right)\left(x+4\right)$$

Combine 77x and 77x to get 154x.

$$154x-308+308=9\left(x-4\right)\left(x+4\right)$$

Add -308 and 308 to get 0.

$$154x=9\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 9 by x-4.

$$154x=\left(9x-36\right)\left(x+4\right)$$

Use the distributive property to multiply 9x-36 by x+4 and combine like terms.

$$154x=9x^{2}-144$$

Subtract 9x^{2} from both sides.

$$154x-9x^{2}=-144$$

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.

$$-9x^{2}+154x=-144$$

Divide both sides by -9.

$$\frac{-9x^{2}+154x}{-9}=-\frac{144}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{154}{-9}x=-\frac{144}{-9}$$

Divide 154 by -9.

$$x^{2}-\frac{154}{9}x=-\frac{144}{-9}$$

Divide -144 by -9.

$$x^{2}-\frac{154}{9}x=16$$

Divide -\frac{154}{9}, the coefficient of the x term, by 2 to get -\frac{77}{9}. Then add the square of -\frac{77}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{154}{9}x+\left(-\frac{77}{9}\right)^{2}=16+\left(-\frac{77}{9}\right)^{2}$$

Square -\frac{77}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{154}{9}x+\frac{5929}{81}=16+\frac{5929}{81}$$

Add 16 to \frac{5929}{81}.

$$x^{2}-\frac{154}{9}x+\frac{5929}{81}=\frac{7225}{81}$$

Factor x^{2}-\frac{154}{9}x+\frac{5929}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{77}{9}\right)^{2}=\frac{7225}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{77}{9}\right)^{2}}=\sqrt{\frac{7225}{81}}$$

Simplify.

$$x-\frac{77}{9}=\frac{85}{9} x-\frac{77}{9}=-\frac{85}{9}$$

Add \frac{77}{9} to both sides of the equation.

$$x=18 x=-\frac{8}{9}$$