Question

Determine the zeros(s) of the quadratic function y=6x2+13x+5
. Show all steps and work to support your answer.

Answer 1

To determine the zeros of the quadratic function \( y = 6x^2 + 13x + 5 \), we need to solve the equation \( 6x^2 + 13x + 5 = 0 \).

### Step 1: Identify the coefficients
The equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 6 \)
- \( b = 13 \)
- \( c = 5 \)

### Step 2: Use the quadratic formula
The quadratic formula is given by:

\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \( D = b^2 - 4ac \):
\[
D = (13)^2 - 4(6)(5)
\]
\[
= 169 - 120
\]
\[
= 49
\]

### Step 4: Substitute values into the quadratic formula
Now that we have the discriminant, we can substitute \( a \), \( b \), and \( D \) into the quadratic formula:
\[
x = \frac{{-13 \pm \sqrt{49}}}{{2 \cdot 6}}
\]
\[
= \frac{{-13 \pm 7}}{{12}}
\]

### Step 5: Calculate the two possible values for \( x \)

1. **First solution**:
\[
x_1 = \frac{{-13 + 7}}{{12}} = \frac{{-6}}{{12}} = -\frac{1}{2}
\]

2. **Second solution**:
\[
x_2 = \frac{{-13 - 7}}{{12}} = \frac{{-20}}{{12}} = -\frac{5}{3}
\]

### Conclusion
The zeros of the quadratic function \( y = 6x^2 + 13x + 5 \) are:
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]