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Solving the Equation: (x^(2)+3x)(2x+3)-16(2x+3)/(x^(2)+3x) =0
This equation presents a challenge due to its fractional form and the presence of a common factor. Let's break down the steps to solve it:
1. Factor out the Common Factor:
Notice that (2x+3) appears in both terms of the equation. Let's factor it out:
(2x+3) * [(x^(2)+3x) - 16/(x^(2)+3x)] = 0
2. Simplify the Expression Inside the Brackets:
To simplify further, let's get a common denominator for the terms inside the brackets:
(2x+3) * [(x^(2)+3x)(x^(2)+3x)/ (x^(2)+3x) - 16/(x^(2)+3x)] = 0
Now, we can combine the numerators:
(2x+3) * [(x^(4) + 6x^(3) + 9x^(2) - 16)/(x^(2)+3x)] = 0
3. Solve for Possible Values of x:
For the entire expression to equal zero, at least one of the factors must be zero. This gives us two possibilities:
a) (2x+3) = 0
Solving for x, we get: x = -3/2
b) (x^(4) + 6x^(3) + 9x^(2) - 16) = 0
This is a quartic equation, which can be difficult to solve directly. We can explore methods like factoring by grouping or using numerical methods to find approximate solutions.
4. Checking for Validity:
It is crucial to check if the solutions we find are valid. We need to ensure that the denominator (x^(2)+3x) is not zero for any of the solutions.
If we substitute x = -3/2 into the denominator, we get:
(-3/2)^(2) + 3 * (-3/2) = 9/4 - 9/2 = -9/4
This is not equal to zero, so x = -3/2 is a valid solution.
For the solutions obtained from the quartic equation, we need to ensure that the denominator is not zero as well.
Conclusion:
The equation (x^(2)+3x)(2x+3)-16(2x+3)/(x^(2)+3x) =0 has at least one solution: x = -3/2. It may have other solutions depending on the solutions of the quartic equation. Remember to check the validity of all solutions by ensuring the denominator is not zero.