Question

Determine the zero(s) of the quadratic function y=6x2+13x+5 . Show all steps.(2 points)

Answer 1

To find the zeros of the quadratic function \( y = 6x^2 + 13x + 5 \), we need to solve the equation \( 6x^2 + 13x + 5 = 0 \). We can do this using the quadratic formula:

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

where \( a = 6 \), \( b = 13 \), and \( c = 5 \).

### Step 1: Calculate the discriminant

First, we calculate the discriminant \( D = b^2 - 4ac \):

\[
D = 13^2 - 4(6)(5)
\]
\[
D = 169 - 120
\]
\[
D = 49
\]

### Step 2: Apply the quadratic formula

Now, we can apply the values into the quadratic formula:

\[
x = \frac{-13 \pm \sqrt{49}}{2(6)}
\]

### Step 3: Simplify the square root

Since \( \sqrt{49} = 7 \), we can substitute this into the formula:

\[
x = \frac{-13 \pm 7}{12}
\]

### Step 4: Calculate the two possible values of \( x \)

Now we will calculate the two possible solutions:

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:

\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]