To find the roots and the vertex of the quadratic function \( y = -x^2 - 9x + 79 \), we'll start by using the quadratic formula for finding the roots and then use the vertex formula to find the vertex.
Step 1: Find the roots using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \( y = -x^2 - 9x + 79 \):
- \( a = -1 \)
- \( b = -9 \)
- \( c = 79 \)
Substituting these values into the formula:
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Calculate the discriminant: \[ b^2 - 4ac = (-9)^2 - 4(-1)(79) = 81 + 316 = 397 \]
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Use the quadratic formula: \[ x = \frac{-(-9) \pm \sqrt{397}}{2(-1)} = \frac{9 \pm \sqrt{397}}{-2} \]
Calculate \( \sqrt{397} \) (approximately 19.93), then:
\[ x_1 = \frac{9 + 19.93}{-2} \quad \text{and} \quad x_2 = \frac{9 - 19.93}{-2} \]
Calculating the above:
- \( x_1 = \frac{28.93}{-2} \approx -14.465 \)
- \( x_2 = \frac{-10.93}{-2} \approx 5.465 \)
So the roots rounded to three decimal places are:
\[ x_1 \approx -14.465, \quad x_2 \approx 5.465 \]
Step 2: Find the vertex of the quadratic
The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting \( a = -1 \) and \( b = -9 \):
\[ x = -\frac{-9}{2 \cdot -1} = \frac{9}{-2} = -4.5 \]
Next, we find the corresponding y-coordinate by substituting \( x = -4.5 \) back into the original equation:
\[ y = -(-4.5)^2 - 9(-4.5) + 79 \] \[ y = -20.25 + 40.5 + 79 \] \[ y = 99.25 \]
Thus, the vertex of the quadratic rounded to three decimal places is:
\[ \text{Vertex} \approx (-4.500, 99.250) \]
Final Summary
- Roots: \( x_1 \approx -14.465, \quad x_2 \approx 5.465 \)
- Vertex: \( (-4.500, 99.250) \)