Finding the Difference Between Roots
We are asked to find the difference between the roots of the quadratic equation \(x^2 - 6x - 16 = 0\).
This equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 1\)
- \(b = -6\)
- \(c = -16\)
Calculate Root Difference Using Formula
The difference between the roots (\(\alpha\) and \(\beta\)) of a quadratic equation can be calculated using the formula:
\(|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}\)
First, calculate the discriminant, \(\Delta = b^2 - 4ac\): \( \Delta = (-6)^2 - 4(1)(-16) \) \( \Delta = 36 + 64 \) \( \Delta = 100 \) Now, substitute the values into the difference formula:
\(|\alpha - \beta| = \frac{\sqrt{100}}{|1|}\) \(|\alpha - \beta| = \frac{10}{1}\) \(|\alpha - \beta| = 10\) Therefore, the difference between the roots of the equation \(x^2 - 6x - 16 = 0\) is 10.
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