Solve quadtext{17.}x-3/x+3+x+3/x-3=21/2

$$\quad \text { 17. } \frac { x - 3 } { x + 3 } + \frac { x + 3 } { x - 3 } = 2 \frac { 1 } { 2 }$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{247}i+96}{31}\approx 3.096774194-1.520925837i$$

$$x=\frac{96+3\sqrt{247}i}{31}\approx 3.096774194+1.520925837i$$

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 2\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3,2.

$$17\left(2x-6\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 17 by 2x-6.

$$\left(34x-102\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 34x-102 by x-3 and combine like terms.

$$34x^{2}-204x+306+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x+6 by x+3 and combine like terms.

$$34x^{2}-204x+306+2x^{2}+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 34x^{2} and 2x^{2} to get 36x^{2}.

$$36x^{2}-204x+306+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine -204x and 12x to get -192x.

$$36x^{2}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Add 306 and 18 to get 324.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Multiply 2 and 2 to get 4.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(4+1\right)$$

Add 4 and 1 to get 5.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\times 5$$

Use the distributive property to multiply x^{2}-9 by 5.

$$36x^{2}-192x+324=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$36x^{2}-192x+324-5x^{2}=-45$$

Combine 36x^{2} and -5x^{2} to get 31x^{2}.

$$31x^{2}-192x+324=-45$$

Add 45 to both sides.

$$31x^{2}-192x+324+45=0$$

Add 324 and 45 to get 369.

$$31x^{2}-192x+369=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 31 for a, -192 for b, and 369 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-192\right)±\sqrt{\left(-192\right)^{2}-4\times 31\times 369}}{2\times 31}$$

Square -192.

$$x=\frac{-\left(-192\right)±\sqrt{36864-4\times 31\times 369}}{2\times 31}$$

Multiply -4 times 31.

$$x=\frac{-\left(-192\right)±\sqrt{36864-124\times 369}}{2\times 31}$$

Multiply -124 times 369.

$$x=\frac{-\left(-192\right)±\sqrt{36864-45756}}{2\times 31}$$

Add 36864 to -45756.

$$x=\frac{-\left(-192\right)±\sqrt{-8892}}{2\times 31}$$

Take the square root of -8892.

$$x=\frac{-\left(-192\right)±6\sqrt{247}i}{2\times 31}$$

The opposite of -192 is 192.

$$x=\frac{192±6\sqrt{247}i}{2\times 31}$$

Multiply 2 times 31.

$$x=\frac{192±6\sqrt{247}i}{62}$$

Now solve the equation x=\frac{192±6\sqrt{247}i}{62} when ± is plus. Add 192 to 6i\sqrt{247}.

$$x=\frac{192+6\sqrt{247}i}{62}$$

Divide 192+6i\sqrt{247} by 62.

$$x=\frac{96+3\sqrt{247}i}{31}$$

Now solve the equation x=\frac{192±6\sqrt{247}i}{62} when ± is minus. Subtract 6i\sqrt{247} from 192.

$$x=\frac{-6\sqrt{247}i+192}{62}$$

Divide 192-6i\sqrt{247} by 62.

$$x=\frac{-3\sqrt{247}i+96}{31}$$

The equation is now solved.

$$x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}$$

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 2\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3,2.

$$17\left(2x-6\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 17 by 2x-6.

$$\left(34x-102\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 34x-102 by x-3 and combine like terms.

$$34x^{2}-204x+306+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x+6 by x+3 and combine like terms.

$$34x^{2}-204x+306+2x^{2}+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 34x^{2} and 2x^{2} to get 36x^{2}.

$$36x^{2}-204x+306+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine -204x and 12x to get -192x.

$$36x^{2}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Add 306 and 18 to get 324.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Multiply 2 and 2 to get 4.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(4+1\right)$$

Add 4 and 1 to get 5.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\times 5$$

Use the distributive property to multiply x^{2}-9 by 5.

$$36x^{2}-192x+324=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$36x^{2}-192x+324-5x^{2}=-45$$

Combine 36x^{2} and -5x^{2} to get 31x^{2}.

$$31x^{2}-192x+324=-45$$

Subtract 324 from both sides.

$$31x^{2}-192x=-45-324$$

Subtract 324 from -45 to get -369.

$$31x^{2}-192x=-369$$

Divide both sides by 31.

$$\frac{31x^{2}-192x}{31}=-\frac{369}{31}$$

Dividing by 31 undoes the multiplication by 31.

$$x^{2}-\frac{192}{31}x=-\frac{369}{31}$$

Divide -\frac{192}{31}, the coefficient of the x term, by 2 to get -\frac{96}{31}. Then add the square of -\frac{96}{31} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{192}{31}x+\left(-\frac{96}{31}\right)^{2}=-\frac{369}{31}+\left(-\frac{96}{31}\right)^{2}$$

Square -\frac{96}{31} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{192}{31}x+\frac{9216}{961}=-\frac{369}{31}+\frac{9216}{961}$$

Add -\frac{369}{31} to \frac{9216}{961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{192}{31}x+\frac{9216}{961}=-\frac{2223}{961}$$

Factor x^{2}-\frac{192}{31}x+\frac{9216}{961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{96}{31}\right)^{2}=-\frac{2223}{961}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{96}{31}\right)^{2}}=\sqrt{-\frac{2223}{961}}$$

Simplify.

$$x-\frac{96}{31}=\frac{3\sqrt{247}i}{31} x-\frac{96}{31}=-\frac{3\sqrt{247}i}{31}$$

Add \frac{96}{31} to both sides of the equation.

$$x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}$$