Question Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary). $$ y=-x^{2}+8 x+65 $$ Answer Attempt 1 out of 2 Roots: $$ \square $$ and

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To find the **roots** and the **vertex** of the quadratic equation $$ y = -x^{2} + 8x + 65 $$, we can follow these steps:

### 1. Finding the Roots
The roots of the quadratic equation are the values of $$ x $$ for which $$ y = 0 $$.

Set $$ y = 0 $$:
$$
- x^{2} + 8x + 65 = 0
$$

Multiply the entire equation by $$-1$$ to make it easier to work with:
$$
x^{2} - 8x - 65 = 0
$$

Use the **quadratic formula**:
$$
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$
where $$ a = 1 $$, $$ b = -8 $$, and $$ c = -65 $$.

Calculate the discriminant ($$ \Delta $$):
$$
\Delta = b^{2} - 4ac = (-8)^{2} - 4(1)(-65) = 64 + 260 = 324
$$

Since the discriminant is positive and a perfect square, there are two real roots:
$$
x = \frac{8 \pm \sqrt{324}}{2} = \frac{8 \pm 18}{2}
$$

Calculate the two roots:
$$
x_{1} = \frac{8 + 18}{2} = \frac{26}{2} = 13
$$
$$
x_{2} = \frac{8 - 18}{2} = \frac{-10}{2} = -5
$$

**Roots:** $$ x = -5 $$ and $$ x = 13 $$

### 2. Finding the Vertex
The vertex of a quadratic equation $$ y = ax^{2} + bx + c $$ is given by:
$$
\left( h, k \right) = \left( -\frac{b}{2a}, y\left( -\frac{b}{2a} \right) \right)
$$

Given $$ a = -1 $$ and $$ b = 8 $$:
$$
h = -\frac{8}{2(-1)} = 4
$$

Calculate $$ k $$ by substituting $$ h = 4 $$ back into the equation:
$$
k = - (4)^{2} + 8(4) + 65 = -16 + 32 + 65 = 81
$$

**Vertex:** $$ (4,\ 81) $$

### **Final Answer**
- **Roots:** $$ -5 $$ and $$ 13 $$
- **Vertex:** $$ (4,\ 81) $$
**Roots:** $$ x = -5 $$ and $$ x = 13 $$ **Vertex:** $$ (4,\ 81) $$

Question Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary). \[ y=-x^{2}+8 x+65 \] Answer Attempt 1 out of 2 Roots: \( \square \) and

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