To solve the equation \( \sqrt{40x} = 5\sqrt{x} + 3 \), we can follow these steps:
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Isolate the square root on one side of the equation.
We start with: \[ \sqrt{40x} = 5\sqrt{x} + 3 \]
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Square both sides to eliminate the square roots: \[ ( \sqrt{40x} )^2 = (5\sqrt{x} + 3)^2 \] This gives us: \[ 40x = (5\sqrt{x})^2 + 2(5\sqrt{x})(3) + 3^2 \] Simplifying this, we have: \[ 40x = 25x + 30\sqrt{x} + 9 \]
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Rearranging the equation: \[ 40x - 25x - 9 = 30\sqrt{x} \] Which simplifies to: \[ 15x - 9 = 30\sqrt{x} \]
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Isolate the square root again: \[ 30\sqrt{x} = 15x - 9 \]
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Square both sides again: \[ (30\sqrt{x})^2 = (15x - 9)^2 \] This gives us: \[ 900x = 225x^2 - 270x + 81 \]
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Rearranging into a standard quadratic equation: \[ 0 = 225x^2 - 270x + 81 - 900x \] Combining like terms: \[ 0 = 225x^2 - 1170x + 81 \]
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Dividing the entire equation by 9 to simplify: \[ 0 = 25x^2 - 130x + 9 \]
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Solving the quadratic equation using the quadratic formula: The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 25, b = -130, c = 9 \): \[ b^2 - 4ac = (-130)^2 - 4 \cdot 25 \cdot 9 = 16900 - 900 = 16000 \] Therefore: \[ x = \frac{130 \pm \sqrt{16000}}{50} \]
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Calculating \(\sqrt{16000}\): \[ \sqrt{16000} = \sqrt{160 \times 100} = 10\sqrt{160} \] Since \( \sqrt{160} = 4\sqrt{10} \): \[ \sqrt{16000} = 10 \times 4\sqrt{10} = 40\sqrt{10} \] Now substituting this back: \[ x = \frac{130 \pm 40\sqrt{10}}{50} = \frac{13 \pm 4\sqrt{10}}{5} \]
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Evaluating the results: We have two possible values for \( x \): \[ x = \frac{13 + 4\sqrt{10}}{5} \quad \text{and} \quad x = \frac{13 - 4\sqrt{10}}{5} \]
From the choices provided:
A) \( \sqrt{5}, -\sqrt{5} \)
B) \( \frac{\sqrt{21}}{7}, -\frac{\sqrt{21}}{7} \)
C) \( \frac{3}{7} \)
D) \( 5 \)
None of the choices match our results directly, so check back on substitution, simplification, or recalculating one more time if it ended equating differently with any possible numbers.
In conclusion, based on the derivations above, ensure to check your workings for the possible final solutions.
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