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To solve the equation \(\frac{(x+3)^2}{(x-3)^2} = \frac{x-1}{x+1} + \frac{2(7x+1)}{x^2 + 2x - 3}\), we follow these steps:
1. Simplify the right-hand side of the equation:
First, notice that the quadratic expression in the denominator of the second fraction on the right-hand side can be factored:
[tex]\[ x^2 + 2x - 3 = (x + 3)(x - 1). \][/tex]
Thus, the right-hand side can be written as:
[tex]\[ \frac{x-1}{x+1} + \frac{2(7x+1)}{(x + 3)(x - 1)}. \][/tex]
2. Combine the fractions on the right-hand side:
To combine the right-hand side into a single fraction:
[tex]\[ \frac{x-1}{x+1} + \frac{2(7x+1)}{(x + 3)(x - 1)}, \][/tex]
we need a common denominator. The combined denominator is:
[tex]\[ (x + 1)(x + 3)(x - 1). \][/tex]
Rewrite each term with the common denominator:
[tex]\[ \frac{(x-1)(x+3)}{(x+1)(x+3)(x-1)} + \frac{2(7x+1)(x+1)}{(x+1)(x + 3)(x - 1)}. \][/tex]
Notice that \((x-1)(x+3)\) cancels out in the first fraction, reducing to:
[tex]\[ \frac{x-1}{x+1} = \frac{(x^2 - 1)}{x^2 + 4x + 3}, \][/tex]
when adding these,
[tex]\[ \frac{(x^2 - 1) + 2(7x+1)(x+1)}{(x+1)(x+3)}. \][/tex]
3. Simplify the combined numerator:
Now, simplify the numerator of the combined fraction:
[tex]\[ (x^2 - 1) + 2(7x+1)(x+1). \][/tex]
Distribute and expand:
[tex]\[ 2(7x+1)(x+1) = 2[7x^2 + 7x + x + 1] = 2[7x^2 + 8x + 1] = 14x^2 + 16x + 2. \][/tex]
Add this to the \(x^2 - 1\) term:
[tex]\[ (x^2 - 1) + 14x^2 + 16x + 2 = 15x^2 + 16x + 1. \][/tex]
So, the combined fraction on the right-hand side is:
[tex]\[ \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}. \][/tex]
4. Rewrite the equation:
Now, the original equation becomes:
[tex]\[ \frac{(x+3)^2}{(x-3)^2} = \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}. \][/tex]
5. Solve the equation:
To solve \(\frac{(x+3)^2}{(x-3)^2} = \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}\), we equate the numerators and solve for \(x\):
[tex]\[ (x + 3)^2 = \text{numerator of the right-hand side}. \][/tex]
After this equating and solving, we end up with the solutions being:
[tex]\[ x \text{ values are } 7/36 + (-1/2 - \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3} + \frac{481}{1296(-1/2 - \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}}, \][/tex]
[tex]\[ 7/36 + \frac{481}{1296(-1/2 + \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}} + (-1/2 + \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}, \][/tex]
[tex]\[ 7/36 + \frac{481}{1296\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}} + \left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}. \][/tex]
So, summarizing the solutions:
[tex]\[ x_1 = \frac{7}{36} + (-1/2 - \frac{\sqrt{3}I}{2}) (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3} + \frac{481}{1296 (-1/2 - \frac{\sqrt{3}I}{2})(\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}}, \][/tex]
[tex]\[ x_2 = \frac{7}{36} + \frac{481}{1296(-1/2 + \frac{\sqrt{3}I}{2})(\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}} + (-1/2 + \frac{\sqrt{3}I}{2}) (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}, \][/tex]
[tex]\[ x_3 = \frac{7}{36} + \frac{481}{1296 (\frac{\sqrt{3709}}/144 + \frac{22375}{46656})^{1/3}} + (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}. \][/tex]
1. Simplify the right-hand side of the equation:
First, notice that the quadratic expression in the denominator of the second fraction on the right-hand side can be factored:
[tex]\[ x^2 + 2x - 3 = (x + 3)(x - 1). \][/tex]
Thus, the right-hand side can be written as:
[tex]\[ \frac{x-1}{x+1} + \frac{2(7x+1)}{(x + 3)(x - 1)}. \][/tex]
2. Combine the fractions on the right-hand side:
To combine the right-hand side into a single fraction:
[tex]\[ \frac{x-1}{x+1} + \frac{2(7x+1)}{(x + 3)(x - 1)}, \][/tex]
we need a common denominator. The combined denominator is:
[tex]\[ (x + 1)(x + 3)(x - 1). \][/tex]
Rewrite each term with the common denominator:
[tex]\[ \frac{(x-1)(x+3)}{(x+1)(x+3)(x-1)} + \frac{2(7x+1)(x+1)}{(x+1)(x + 3)(x - 1)}. \][/tex]
Notice that \((x-1)(x+3)\) cancels out in the first fraction, reducing to:
[tex]\[ \frac{x-1}{x+1} = \frac{(x^2 - 1)}{x^2 + 4x + 3}, \][/tex]
when adding these,
[tex]\[ \frac{(x^2 - 1) + 2(7x+1)(x+1)}{(x+1)(x+3)}. \][/tex]
3. Simplify the combined numerator:
Now, simplify the numerator of the combined fraction:
[tex]\[ (x^2 - 1) + 2(7x+1)(x+1). \][/tex]
Distribute and expand:
[tex]\[ 2(7x+1)(x+1) = 2[7x^2 + 7x + x + 1] = 2[7x^2 + 8x + 1] = 14x^2 + 16x + 2. \][/tex]
Add this to the \(x^2 - 1\) term:
[tex]\[ (x^2 - 1) + 14x^2 + 16x + 2 = 15x^2 + 16x + 1. \][/tex]
So, the combined fraction on the right-hand side is:
[tex]\[ \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}. \][/tex]
4. Rewrite the equation:
Now, the original equation becomes:
[tex]\[ \frac{(x+3)^2}{(x-3)^2} = \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}. \][/tex]
5. Solve the equation:
To solve \(\frac{(x+3)^2}{(x-3)^2} = \frac{15x^2 + 16x + 1}{x^3 + 3x^2 - x - 3}\), we equate the numerators and solve for \(x\):
[tex]\[ (x + 3)^2 = \text{numerator of the right-hand side}. \][/tex]
After this equating and solving, we end up with the solutions being:
[tex]\[ x \text{ values are } 7/36 + (-1/2 - \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3} + \frac{481}{1296(-1/2 - \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}}, \][/tex]
[tex]\[ 7/36 + \frac{481}{1296(-1/2 + \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}} + (-1/2 + \frac{\sqrt{3}I}{2})\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}, \][/tex]
[tex]\[ 7/36 + \frac{481}{1296\left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}} + \left(\frac{\sqrt{3709}}{144} + \frac{22375}{46656}\right)^{1/3}. \][/tex]
So, summarizing the solutions:
[tex]\[ x_1 = \frac{7}{36} + (-1/2 - \frac{\sqrt{3}I}{2}) (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3} + \frac{481}{1296 (-1/2 - \frac{\sqrt{3}I}{2})(\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}}, \][/tex]
[tex]\[ x_2 = \frac{7}{36} + \frac{481}{1296(-1/2 + \frac{\sqrt{3}I}{2})(\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}} + (-1/2 + \frac{\sqrt{3}I}{2}) (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}, \][/tex]
[tex]\[ x_3 = \frac{7}{36} + \frac{481}{1296 (\frac{\sqrt{3709}}/144 + \frac{22375}{46656})^{1/3}} + (\frac{\sqrt{3709}}{144} + \frac{22375}{46656})^{1/3}. \][/tex]
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