To solve the equation \( \frac{14}{4a + 16} = -a + 6 \), follow these steps:
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Cross Multiply: Multiply both sides by \( (4a + 16) \) to eliminate the fraction: \[ 14 = (-a + 6)(4a + 16) \]
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Expand the Right Side: Distributing \(-a + 6\) gives: \[ 14 = -4a^2 - 16a + 24a + 96 \] Which simplifies to: \[ 14 = -4a^2 + 8a + 96 \]
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Rearranging the Equation: Move everything to one side to set it to 0: \[ 0 = -4a^2 + 8a + 96 - 14 \] This simplifies to: \[ 0 = -4a^2 + 8a + 82 \] To make it cleaner, multiply through by \(-1\): \[ 0 = 4a^2 - 8a - 82 \]
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Quadratic Formula: Use the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 4\), \(b = -8\), and \(c = -82\): \[ a = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(-82)}}{2(4)} \] \[ a = \frac{8 \pm \sqrt{64 + 1312}}{8} \] \[ a = \frac{8 \pm \sqrt{1376}}{8} \] \[ a = \frac{8 \pm 4\sqrt{86}}{8} \] \[ a = 1 \pm \frac{\sqrt{86}}{2} \]
The two potential solutions are: \[ a = 1 + \frac{\sqrt{86}}{2} \quad \text{or} \quad a = 1 - \frac{\sqrt{86}}{2} \]
To check which (if either) fits the original equation, plug them back into \( \frac{14}{4a + 16} = -a + 6 \) and verify if both sides are equal.
For numeric approximation:
- Calculate \( \sqrt{86} \approx 9.273 \), hence: \[ a \approx 1 + 4.6365 \quad \text{or} \quad a \approx 1 - 4.6365 \] This results in about: \[ a \approx 5.6365 \quad \text{or} \quad a \approx -3.6365 \]
You can plug these approximate values back into the original equation.
Choose any one of the values (or both) to verify by substitution into the original equation and confirming if both sides are equal.
Let me know if you would like help finding a specific answer or verification!