\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
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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)
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.
\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 17 by 2x-6.
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 34x-102 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)
Use the distributive property to multiply 2x+6 by x+3 and combine like terms.
36x^{2}-204x+306+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}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)
Combine -204x and 12x to get -192x.
36x^{2}-192x+324=\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(4+1\right)
Multiply 2 and 2 to get 4.
36x^{2}-192x+324=\left(x^{2}-9\right)\times 5
Add 4 and 1 to get 5.
36x^{2}-192x+324=5x^{2}-45
Use the distributive property to multiply x^{2}-9 by 5.
36x^{2}-192x+324-5x^{2}=-45
Subtract 5x^{2} from both sides.
31x^{2}-192x+324=-45
Combine 36x^{2} and -5x^{2} to get 31x^{2}.
31x^{2}-192x+324+45=0
Add 45 to both sides.
31x^{2}-192x+369=0
Add 324 and 45 to get 369.
x=\frac{-\left(-192\right)±\sqrt{\left(-192\right)^{2}-4\times 31\times 369}}{2\times 31}
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{36864-4\times 31\times 369}}{2\times 31}
Square -192.
x=\frac{-\left(-192\right)±\sqrt{36864-124\times 369}}{2\times 31}
Multiply -4 times 31.
x=\frac{-\left(-192\right)±\sqrt{36864-45756}}{2\times 31}
Multiply -124 times 369.
x=\frac{-\left(-192\right)±\sqrt{-8892}}{2\times 31}
Add 36864 to -45756.
x=\frac{-\left(-192\right)±6\sqrt{247}i}{2\times 31}
Take the square root of -8892.
x=\frac{192±6\sqrt{247}i}{2\times 31}
The opposite of -192 is 192.
x=\frac{192±6\sqrt{247}i}{62}
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{96+3\sqrt{247}i}{31}
Divide 192+6i\sqrt{247} by 62.
x=\frac{-6\sqrt{247}i+192}{62}
Now solve the equation x=\frac{192±6\sqrt{247}i}{62} when ± is minus. Subtract 6i\sqrt{247} from 192.
x=\frac{-3\sqrt{247}i+96}{31}
Divide 192-6i\sqrt{247} by 62.
x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}
The equation is now solved.
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)
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.
\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 17 by 2x-6.
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 34x-102 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)
Use the distributive property to multiply 2x+6 by x+3 and combine like terms.
36x^{2}-204x+306+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}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)
Combine -204x and 12x to get -192x.
36x^{2}-192x+324=\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(4+1\right)
Multiply 2 and 2 to get 4.
36x^{2}-192x+324=\left(x^{2}-9\right)\times 5
Add 4 and 1 to get 5.
36x^{2}-192x+324=5x^{2}-45
Use the distributive property to multiply x^{2}-9 by 5.
36x^{2}-192x+324-5x^{2}=-45
Subtract 5x^{2} from both sides.
31x^{2}-192x+324=-45
Combine 36x^{2} and -5x^{2} to get 31x^{2}.
31x^{2}-192x=-45-324
Subtract 324 from both sides.
31x^{2}-192x=-369
Subtract 324 from -45 to get -369.
\frac{31x^{2}-192x}{31}=-\frac{369}{31}
Divide both sides by 31.
x^{2}-\frac{192}{31}x=-\frac{369}{31}
Dividing by 31 undoes the multiplication by 31.
x^{2}-\frac{192}{31}x+\left(-\frac{96}{31}\right)^{2}=-\frac{369}{31}+\left(-\frac{96}{31}\right)^{2}
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+\frac{9216}{961}=-\frac{369}{31}+\frac{9216}{961}
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{2223}{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.
\left(x-\frac{96}{31}\right)^{2}=-\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}.
\sqrt{\left(x-\frac{96}{31}\right)^{2}}=\sqrt{-\frac{2223}{961}}
Take the square root of both sides of the equation.
x-\frac{96}{31}=\frac{3\sqrt{247}i}{31} x-\frac{96}{31}=-\frac{3\sqrt{247}i}{31}
Simplify.
x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}
Add \frac{96}{31} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}