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The solution of the equation x/(x-1) + 2/(x+2) = -6/(x²+x-2) represented by statement is option D. The equation has one true solution and one extraneous solution.
To find the solutions to the equation, we need to simplify and solve for x:
x/(x-1) + 2/(x+2) = -6/(x²+x-2)
Factor the denominators on both sides of the equation.
The denominator on the left side can be factored as follows:
x² + x - 2 = (x+2)(x-1)
Find the common denominator.
The common denominator for the left side is (x-1)(x+2).
Rewrite the fractions with the common denominator:
[(x+2) × x]/[(x-1)(x+2)] + [2 × (x-1)]/[(x-1)(x+2)] = -6/[(x+2)(x-1)]
Combine the fractions,
[x(x+2) + 2(x-1)] / [(x-1)(x+2)] = -6/[(x+2)(x-1)]
Eliminate the denominator:
Multiply both sides by (x+2)(x-1) to eliminate the denominators:
[x(x+2) + 2(x-1)] = -6
Expand and simplify the equation:
x² + 2x + 2x - 2 = -6
x² + 4x - 2 = -6
x² + 4x + 4 = 0
Solve for x using the quadratic formula:
x = (-b ± √(b²-4ac)) / 2a
Here, a = 1, b = 4, and c = 4.
x = (-4 ± √(4² - 4 × 1 × 4)) / 2× 1
x = (-4 ± √(16 - 16)) / 2
x = (-4 ± √0) / 2
x = (-4 ± 0) / 2
x = -4 / 2
x = -2
The equation has one true solution, x = -2 (repeated root).
Now, let's check this solution in the original equation:
x/(x-1) + 2/(x+2) = -6/(x²+x-2)
(-2)/((-2)-1) + 2/((-2)+2) = -6/((-2)²+(-2)-2)
(-2)/(-3) + 2/0 = -6/(4-2-2)
2/0 is undefined, and since we cannot divide by zero, we have an extraneous solution.
Therefore, the correct statement for the solution is option D. The equation has one true solution and one extraneous solution.
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