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(3)/(x)-(2)/(x-3)=(12)/(9-x^(r))

$\frac{3}{x}-\frac{2}{x-3}=\frac{12}{9-x^{r}}$

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Sum of finite series starts from 1

Full solution

Q.

\frac{3}{x}-\frac{2}{x-3}=\frac{12}{9-x^{r}}

Combine Fractions: Combine the fractions on the left side by finding a common denominator, which is $x(x-3)$ .
Simplify Equation: $(3(x-3)-2x)/(x(x-3))=(12)/(9-x^{2})$ .
Factorize Denominator: Simplify the numerator on the left side: $3x-9-2x$ .
Cross Multiply: This simplifies to $x-9$ in the numerator, so we have $(x-9)/(x(x-3))=(12)/(9-x^{2})$ .
Expand Left Side: Recognize that $9-x^2$ is a difference of squares and can be factored into $(3-x)(3+x)$ .
Simplify Equation: Now we have $(x-9)/(x(x-3))=(12)/((3-x)(3+x))$ .
Combine Like Terms: Cross multiply to get rid of the fractions: x\(-9)( $3$ -x)( $3$ +x)= $12$ x(x $-3$ )\.
Set Equation to Zero: Expand the left side: $(x-9)(9-x^2)=12x^2-36x$ .
Factorize Cubic Equation: Notice that $(x-9)(9-x^2)$ is the same as $-(x-9)(x^2-9)$ because $9-x^2$ is $-(x^2-9)$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $- x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $-x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $-x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $-x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $-x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ . Move all terms to one side to set the equation to zero: $-x^3 + 3x^2 + 6x - 27 - 12x^2 + 36x = 0$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $- x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $- x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ . Move all terms to one side to set the equation to zero: $- x^3 + 3x^2 + 6x - 27 - 12x^2 + 36x = 0$ . Combine like terms: $- x^3 - 9x^2 + 42x - 27 = 0$ .
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $-x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $-x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ . Move all terms to one side to set the equation to zero: $-x^3 + 3x^2 + 6x - 27 - 12x^2 + 36x = 0$ . Combine like terms: $-x^3 - 9x^2 + 42x - 27 = 0$ . This is a cubic equation, and solving for x may require factoring or using the cubic formula. However, we can check for obvious solutions by plugging in values or factoring by grouping.
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $-x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $-x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ . Move all terms to one side to set the equation to zero: $-x^3 + 3x^2 + 6x - 27 - 12x^2 + 36x = 0$ . Combine like terms: $-x^3 - 9x^2 + 42x - 27 = 0$ . This is a cubic equation, and solving for x may require factoring or using the cubic formula. However, we can check for obvious solutions by plugging in values or factoring by grouping. Try to factor by grouping or look for obvious factors.
Check for Mistake: Now we have $- (x - 9)(x + 3)(x - 3) = 12x^2 - 36x$ . Expand the left side: $- (x^3 - 3x^2 + 3x - 9x + 27) = 12x^2 - 36x$ . Simplify the left side: $- x^3 + 3x^2 - 3x + 9x - 27 = 12x^2 - 36x$ . Combine like terms on the left side: $- x^3 + 3x^2 + 6x - 27 = 12x^2 - 36x$ . Move all terms to one side to set the equation to zero: $- x^3 + 3x^2 + 6x - 27 - 12x^2 + 36x = 0$ . Combine like terms: $- x^3 - 9x^2 + 42x - 27 = 0$ . This is a cubic equation, and solving for x may require factoring or using the cubic formula. However, we can check for obvious solutions by plugging in values or factoring by grouping. Try to factor by grouping or look for obvious factors. Notice that there is no obvious way to factor this cubic equation, and it seems like we might have made a mistake. Let's go back and check our steps.

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