Question

Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary).
y, equals, 20, x, squared, plus, 180, x, minus, 567
y=20x
2
+180x−567

Answer 1

To find the roots and the vertex of the quadratic function \( y = 20x^2 + 180x - 567 \), we can use the following formulas:

1. **Roots:** To find the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In this case, \( a = 20 \), \( b = 180 \), and \( c = -567 \).

2. **Vertex:** The vertex \( (h, k) \) of a quadratic given in standard form \( ax^2 + bx + c \) can be found using:
\[
h = -\frac{b}{2a}
\]
and then substituting \( h \) back into the equation to find \( k \).

Let's calculate:

### Step 1: Calculate the roots

#### Using the Quadratic Formula:

1. **Calculate \( b^2 - 4ac \)**:

\[
b^2 - 4ac = 180^2 - 4(20)(-567)
\]
\[
= 32400 + 45360 = 77760
\]

2. **Calculate the roots**:

\[
x = \frac{-180 \pm \sqrt{77760}}{2(20)}
\]
\[
= \frac{-180 \pm 278.538}{40}
\]

- **First root:**

\[
x_1 = \frac{-180 + 278.538}{40} = \frac{98.538}{40} \approx 2.463
\]

- **Second root:**

\[
x_2 = \frac{-180 - 278.538}{40} = \frac{-458.538}{40} \approx -11.463
\]

### Roots:
- \( x_1 \approx 2.463 \)
- \( x_2 \approx -11.463 \)

### Step 2: Calculate the vertex

1. **Calculate \( h \)**:

\[
h = -\frac{180}{2 \times 20} = -\frac{180}{40} = -4.5
\]

2. **Calculate \( k \)** by substituting \( h \) back into the original equation:

\[
k = 20(-4.5)^2 + 180(-4.5) - 567
\]
\[
= 20(20.25) - 810 - 567
\]
\[
= 405 - 810 - 567 = -972
\]

### Vertex:
- \( (h, k) = (-4.5, -972) \)

### Summary:
- **Roots:** \( x_1 \approx 2.463, x_2 \approx -11.463 \)
- **Vertex:** \( (-4.5, -972) \)