To solve the equation \(-5k^2 + 4 = -8k\) using the quadratic formula, we first rewrite the equation in standard form \(ax^2 + bx + c = 0\).
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Rearranging the equation: \[ -5k^2 + 8k + 4 = 0 \]
We can multiply the entire equation by \(-1\) to make the leading coefficient positive: \[ 5k^2 - 8k - 4 = 0 \]
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Identifying coefficients: Here, \(a = 5\), \(b = -8\), and \(c = -4\).
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Using the quadratic formula: The quadratic formula is given by: \[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Plugging in the values of \(a\), \(b\), and \(c\):
- First calculate \(b^2 - 4ac\): \[ b^2 = (-8)^2 = 64 \] \[ 4ac = 4 \times 5 \times (-4) = -80 \] \[ b^2 - 4ac = 64 - (-80) = 64 + 80 = 144 \]
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Calculating the roots: Now substitute back into the quadratic formula: \[ k = \frac{-(-8) \pm \sqrt{144}}{2 \cdot 5} \] \[ k = \frac{8 \pm 12}{10} \]
This gives us two solutions:
- For \(k = \frac{8 + 12}{10}\): \[ k = \frac{20}{10} = 2 \]
- For \(k = \frac{8 - 12}{10}\): \[ k = \frac{-4}{10} = -\frac{2}{5} \]
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Final solutions: Therefore, the solutions are: \[ k = 2 \quad \text{and} \quad k = -\frac{2}{5} \]
The solution presented in your message appears to be incorrect. The true solutions are \(\mathbf{k = 2}\) and \(\mathbf{k = -\frac{2}{5}}\).