Solve 3(x-3/x-2)+168(x-3/x+2)-46(x^2-4)+x-9/x^2-4=00

$$3( \frac{ x-3 }{ x-2 } )+168( \frac{ x-3 }{ x+2 } )- \frac{ 46( { x }^{ 2 } -4)+x-9 }{ { x }^{ 2 } -4 } =00$$

Solve for x

$$x = \frac{\sqrt{30209} + 422}{125} \approx 4.766458917$$

$$x = \frac{422 - \sqrt{30209}}{125} \approx 1.985541083$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,x^{2}-4.

$$3\left(x+2\right)\left(x-3\right)+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 3 by x+2.

$$\left(3x+6\right)\left(x-3\right)+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 3x+6 by x-3 and combine like terms.

$$3x^{2}-3x-18+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 168 by x-2.

$$3x^{2}-3x-18+\left(168x-336\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 168x-336 by x-3 and combine like terms.

$$3x^{2}-3x-18+168x^{2}-840x+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine 3x^{2} and 168x^{2} to get 171x^{2}.

$$171x^{2}-3x-18-840x+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine -3x and -840x to get -843x.

$$171x^{2}-843x-18+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Add -18 and 1008 to get 990.

$$171x^{2}-843x+990-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 46 by x^{2}-4.

$$171x^{2}-843x+990-\left(46x^{2}-184+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Subtract 9 from -184 to get -193.

$$171x^{2}-843x+990-\left(46x^{2}-193+x\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

To find the opposite of 46x^{2}-193+x, find the opposite of each term.

$$171x^{2}-843x+990-46x^{2}+193-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine 171x^{2} and -46x^{2} to get 125x^{2}.

$$125x^{2}-843x+990+193-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Add 990 and 193 to get 1183.

$$125x^{2}-843x+1183-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine -843x and -x to get -844x.

$$125x^{2}-844x+1183=0\times 0\left(x-2\right)\left(x+2\right)$$

Multiply 0 and 0 to get 0.

$$125x^{2}-844x+1183=0\left(x-2\right)\left(x+2\right)$$

Anything times zero gives zero.

$$125x^{2}-844x+1183=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 125 for a, -844 for b, and 1183 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-844\right)±\sqrt{\left(-844\right)^{2}-4\times 125\times 1183}}{2\times 125}$$

Square -844.

$$x=\frac{-\left(-844\right)±\sqrt{712336-4\times 125\times 1183}}{2\times 125}$$

Multiply -4 times 125.

$$x=\frac{-\left(-844\right)±\sqrt{712336-500\times 1183}}{2\times 125}$$

Multiply -500 times 1183.

$$x=\frac{-\left(-844\right)±\sqrt{712336-591500}}{2\times 125}$$

Add 712336 to -591500.

$$x=\frac{-\left(-844\right)±\sqrt{120836}}{2\times 125}$$

Take the square root of 120836.

$$x=\frac{-\left(-844\right)±2\sqrt{30209}}{2\times 125}$$

The opposite of -844 is 844.

$$x=\frac{844±2\sqrt{30209}}{2\times 125}$$

Multiply 2 times 125.

$$x=\frac{844±2\sqrt{30209}}{250}$$

Now solve the equation x=\frac{844±2\sqrt{30209}}{250} when ± is plus. Add 844 to 2\sqrt{30209}.

$$x=\frac{2\sqrt{30209}+844}{250}$$

Divide 844+2\sqrt{30209} by 250.

$$x=\frac{\sqrt{30209}+422}{125}$$

Now solve the equation x=\frac{844±2\sqrt{30209}}{250} when ± is minus. Subtract 2\sqrt{30209} from 844.

$$x=\frac{844-2\sqrt{30209}}{250}$$

Divide 844-2\sqrt{30209} by 250.

$$x=\frac{422-\sqrt{30209}}{125}$$

The equation is now solved.

$$x=\frac{\sqrt{30209}+422}{125} x=\frac{422-\sqrt{30209}}{125}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,x^{2}-4.

$$3\left(x+2\right)\left(x-3\right)+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 3 by x+2.

$$\left(3x+6\right)\left(x-3\right)+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 3x+6 by x-3 and combine like terms.

$$3x^{2}-3x-18+168\left(x-2\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 168 by x-2.

$$3x^{2}-3x-18+\left(168x-336\right)\left(x-3\right)-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 168x-336 by x-3 and combine like terms.

$$3x^{2}-3x-18+168x^{2}-840x+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine 3x^{2} and 168x^{2} to get 171x^{2}.

$$171x^{2}-3x-18-840x+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine -3x and -840x to get -843x.

$$171x^{2}-843x-18+1008-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Add -18 and 1008 to get 990.

$$171x^{2}-843x+990-\left(46\left(x^{2}-4\right)+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 46 by x^{2}-4.

$$171x^{2}-843x+990-\left(46x^{2}-184+x-9\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

Subtract 9 from -184 to get -193.

$$171x^{2}-843x+990-\left(46x^{2}-193+x\right)=0\times 0\left(x-2\right)\left(x+2\right)$$

To find the opposite of 46x^{2}-193+x, find the opposite of each term.

$$171x^{2}-843x+990-46x^{2}+193-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine 171x^{2} and -46x^{2} to get 125x^{2}.

$$125x^{2}-843x+990+193-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Add 990 and 193 to get 1183.

$$125x^{2}-843x+1183-x=0\times 0\left(x-2\right)\left(x+2\right)$$

Combine -843x and -x to get -844x.

$$125x^{2}-844x+1183=0\times 0\left(x-2\right)\left(x+2\right)$$

Multiply 0 and 0 to get 0.

$$125x^{2}-844x+1183=0\left(x-2\right)\left(x+2\right)$$

Anything times zero gives zero.

$$125x^{2}-844x+1183=0$$

Subtract 1183 from both sides. Anything subtracted from zero gives its negation.

$$125x^{2}-844x=-1183$$

Divide both sides by 125.

$$\frac{125x^{2}-844x}{125}=-\frac{1183}{125}$$

Dividing by 125 undoes the multiplication by 125.

$$x^{2}-\frac{844}{125}x=-\frac{1183}{125}$$

Divide -\frac{844}{125}, the coefficient of the x term, by 2 to get -\frac{422}{125}. Then add the square of -\frac{422}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{844}{125}x+\left(-\frac{422}{125}\right)^{2}=-\frac{1183}{125}+\left(-\frac{422}{125}\right)^{2}$$

Square -\frac{422}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{844}{125}x+\frac{178084}{15625}=-\frac{1183}{125}+\frac{178084}{15625}$$

Add -\frac{1183}{125} to \frac{178084}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{844}{125}x+\frac{178084}{15625}=\frac{30209}{15625}$$

Factor x^{2}-\frac{844}{125}x+\frac{178084}{15625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{422}{125}\right)^{2}=\frac{30209}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{422}{125}\right)^{2}}=\sqrt{\frac{30209}{15625}}$$

Simplify.

$$x-\frac{422}{125}=\frac{\sqrt{30209}}{125} x-\frac{422}{125}=-\frac{\sqrt{30209}}{125}$$

Add \frac{422}{125} to both sides of the equation.

$$x=\frac{\sqrt{30209}+422}{125} x=\frac{422-\sqrt{30209}}{125}$$