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$\fraction{9}{x - 3} - \fraction{27}{x (x - 3)} = -3 $
Solve for x
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x\times 9-27=-3x\left(x-3\right)
Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).
x\times 9-27=-3x^{2}+9x
Use the distributive property to multiply -3x by x-3.
x\times 9-27+3x^{2}=9x
Add 3x^{2} to both sides.
x\times 9-27+3x^{2}-9x=0
Subtract 9x from both sides.
-27+3x^{2}=0
Combine x\times 9 and -9x to get 0.
-9+x^{2}=0
Divide both sides by 3.
\left(x-3\right)\left(x+3\right)=0
Consider -9+x^{2}. Rewrite -9+x^{2} as x^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).
x=3 x=-3
To find equation solutions, solve x-3=0 and x+3=0.
x=-3
Variable x cannot be equal to 3.
x\times 9-27=-3x\left(x-3\right)
Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).
x\times 9-27=-3x^{2}+9x
Use the distributive property to multiply -3x by x-3.
x\times 9-27+3x^{2}=9x
Add 3x^{2} to both sides.
x\times 9-27+3x^{2}-9x=0
Subtract 9x from both sides.
-27+3x^{2}=0
Combine x\times 9 and -9x to get 0.
3x^{2}=27
Add 27 to both sides. Anything plus zero gives itself.
x^{2}=\frac{27}{3}
Divide both sides by 3.
x^{2}=9
Divide 27 by 3 to get 9.
x=3 x=-3
Take the square root of both sides of the equation.
x=-3
Variable x cannot be equal to 3.
x\times 9-27=-3x\left(x-3\right)
Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).
x\times 9-27=-3x^{2}+9x
Use the distributive property to multiply -3x by x-3.
x\times 9-27+3x^{2}=9x
Add 3x^{2} to both sides.
x\times 9-27+3x^{2}-9x=0
Subtract 9x from both sides.
-27+3x^{2}=0
Combine x\times 9 and -9x to get 0.
3x^{2}-27=0
Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.
x=\frac{0±\sqrt{0^{2}-4\times 3\left(-27\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{0±\sqrt{-4\times 3\left(-27\right)}}{2\times 3}
Square 0.
x=\frac{0±\sqrt{-12\left(-27\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{0±\sqrt{324}}{2\times 3}
Multiply -12 times -27.
x=\frac{0±18}{2\times 3}
Take the square root of 324.
x=\frac{0±18}{6}
Multiply 2 times 3.
x=3
Now solve the equation x=\frac{0±18}{6} when ± is plus. Divide 18 by 6.
x=-3
Now solve the equation x=\frac{0±18}{6} when ± is minus. Divide -18 by 6.
x=3 x=-3
The equation is now solved.
x=-3
Variable x cannot be equal to 3.