Solve x-3/x+3-x+3/x-3+66/7=0

$$\frac{ x-3 }{ x+3 } - \frac{ x+3 }{ x-3 } +6 \frac{ 6 }{ 7 } =0$$

Solve for x

$$x=4$$

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

Steps Using Factoring By Grouping

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 7\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3,7.

$$\left(7x-21\right)\left(x-3\right)-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x-21 by x-3 and combine like terms.

$$7x^{2}-42x+63-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x+21 by x+3 and combine like terms.

$$7x^{2}-42x+63-\left(7x^{2}+42x+63\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

To find the opposite of 7x^{2}+42x+63, find the opposite of each term.

$$7x^{2}-42x+63-7x^{2}-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-42x+63-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine -42x and -42x to get -84x.

$$-84x+63-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Subtract 63 from 63 to get 0.

$$-84x+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Multiply 6 and 7 to get 42.

$$-84x+\left(x^{2}-9\right)\left(42+6\right)=0$$

Add 42 and 6 to get 48.

$$-84x+\left(x^{2}-9\right)\times 48=0$$

Use the distributive property to multiply x^{2}-9 by 48.

$$-84x+48x^{2}-432=0$$

Divide both sides by 12.

$$-7x+4x^{2}-36=0$$

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

$$4x^{2}-7x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-7 ab=4\left(-36\right)=-144$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Calculate the sum for each pair.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

The solution is the pair that gives sum -7.

$$a=-16 b=9$$

Rewrite 4x^{2}-7x-36 as \left(4x^{2}-16x\right)+\left(9x-36\right).

$$\left(4x^{2}-16x\right)+\left(9x-36\right)$$

Factor out 4x in the first and 9 in the second group.

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

Factor out common term x-4 by using distributive property.

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

To find equation solutions, solve x-4=0 and 4x+9=0.

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

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 7\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3,7.

$$\left(7x-21\right)\left(x-3\right)-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x-21 by x-3 and combine like terms.

$$7x^{2}-42x+63-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x+21 by x+3 and combine like terms.

$$7x^{2}-42x+63-\left(7x^{2}+42x+63\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

To find the opposite of 7x^{2}+42x+63, find the opposite of each term.

$$7x^{2}-42x+63-7x^{2}-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-42x+63-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine -42x and -42x to get -84x.

$$-84x+63-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Subtract 63 from 63 to get 0.

$$-84x+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Multiply 6 and 7 to get 42.

$$-84x+\left(x^{2}-9\right)\left(42+6\right)=0$$

Add 42 and 6 to get 48.

$$-84x+\left(x^{2}-9\right)\times 48=0$$

Use the distributive property to multiply x^{2}-9 by 48.

$$-84x+48x^{2}-432=0$$

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.

$$48x^{2}-84x-432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -84 for b, and -432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 48\left(-432\right)}}{2\times 48}$$

Square -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 48\left(-432\right)}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-84\right)±\sqrt{7056-192\left(-432\right)}}{2\times 48}$$

Multiply -192 times -432.

$$x=\frac{-\left(-84\right)±\sqrt{7056+82944}}{2\times 48}$$

Add 7056 to 82944.

$$x=\frac{-\left(-84\right)±\sqrt{90000}}{2\times 48}$$

Take the square root of 90000.

$$x=\frac{-\left(-84\right)±300}{2\times 48}$$

The opposite of -84 is 84.

$$x=\frac{84±300}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{84±300}{96}$$

Now solve the equation x=\frac{84±300}{96} when ± is plus. Add 84 to 300.

$$x=\frac{384}{96}$$

Divide 384 by 96.

$$x=4$$

Now solve the equation x=\frac{84±300}{96} when ± is minus. Subtract 300 from 84.

$$x=-\frac{216}{96}$$

Reduce the fraction \frac{-216}{96} to lowest terms by extracting and canceling out 24.

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

The equation is now solved.

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

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 7\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3,7.

$$\left(7x-21\right)\left(x-3\right)-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x-21 by x-3 and combine like terms.

$$7x^{2}-42x+63-\left(7x+21\right)\left(x+3\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Use the distributive property to multiply 7x+21 by x+3 and combine like terms.

$$7x^{2}-42x+63-\left(7x^{2}+42x+63\right)+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

To find the opposite of 7x^{2}+42x+63, find the opposite of each term.

$$7x^{2}-42x+63-7x^{2}-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-42x+63-42x-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Combine -42x and -42x to get -84x.

$$-84x+63-63+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Subtract 63 from 63 to get 0.

$$-84x+\left(x^{2}-9\right)\left(6\times 7+6\right)=0$$

Multiply 6 and 7 to get 42.

$$-84x+\left(x^{2}-9\right)\left(42+6\right)=0$$

Add 42 and 6 to get 48.

$$-84x+\left(x^{2}-9\right)\times 48=0$$

Use the distributive property to multiply x^{2}-9 by 48.

$$-84x+48x^{2}-432=0$$

Add 432 to both sides. Anything plus zero gives itself.

$$-84x+48x^{2}=432$$

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.

$$48x^{2}-84x=432$$

Divide both sides by 48.

$$\frac{48x^{2}-84x}{48}=\frac{432}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{84}{48}\right)x=\frac{432}{48}$$

Reduce the fraction \frac{-84}{48} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{7}{4}x=\frac{432}{48}$$

Divide 432 by 48.

$$x^{2}-\frac{7}{4}x=9$$

Divide -\frac{7}{4}, the coefficient of the x term, by 2 to get -\frac{7}{8}. Then add the square of -\frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=9+\left(-\frac{7}{8}\right)^{2}$$

Square -\frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=9+\frac{49}{64}$$

Add 9 to \frac{49}{64}.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{625}{64}$$

Factor x^{2}-\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{625}{64}}$$

Simplify.

$$x-\frac{7}{8}=\frac{25}{8} x-\frac{7}{8}=-\frac{25}{8}$$

Add \frac{7}{8} to both sides of the equation.

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