Question

Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary).
y, equals, minus, x, squared, minus, 9, x, plus, 79
y=−x
2
−9x+79

Answer 1

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:

1. Calculate the discriminant:
\[
b^2 - 4ac = (-9)^2 - 4(-1)(79) = 81 + 316 = 397
\]

2. 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:

1. \( x_1 = \frac{28.93}{-2} \approx -14.465 \)
2. \( x_2 = \frac{-10.93}{-2} \approx 5.465 \)

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:** \( (-4.500, 99.250) \)