Bot GPT 4-o mini To analyze the equation \( (4x^2 - 25)(x^2 - 5x - 14) = 0 \), we can solve each factor separately.
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For the first factor \( 4x^2 - 25 = 0 \):
Rearranging gives: \[ 4x^2 = 25 \quad \Rightarrow \quad x^2 = \frac{25}{4} \] Taking the square root results in: \[ x = \frac{5}{2} \quad \text{or} \quad x = -\frac{5}{2} \] Hence, there are 2 real solutions from this factor.
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For the second factor \( x^2 - 5x - 14 = 0 \):
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -5, \) and \( c = -14 \): \[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2} \] This results in: \[ x = \frac{14}{2} = 7 \quad \text{and} \quad x = \frac{-4}{2} = -2 \] Thus, there are also 2 real solutions from this factor.
In total, combining both factors:
- From \( 4x^2 - 25 = 0 \), we have 2 real solutions.
- From \( x^2 - 5x - 14 = 0 \), we have another 2 real solutions.
Thus, there are a total of 4 real solutions to the original equation.
The correct answer is D. Four real solutions.
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