Question

Determine the zero of the quadratic function y=6x^2+13x+5. Show all steps

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 coefficients
The quadratic 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 calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = 13^2 = 169
\]
\[
4ac = 4 \cdot 6 \cdot 5 = 120
\]
\[
b^2 - 4ac = 169 - 120 = 49
\]

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

We will calculate the two possible values for \( x \):

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

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

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