Question

Find the zeros of the quadratic equation y=6x2+13x+5 . You may use any of the following methods: factoring, square roots, complete the square, or quadratic formula

Answer 1

To find the zeros of the quadratic equation \( y = 6x^2 + 13x + 5 \), we can use the quadratic formula, which is given by:

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

In this equation, \( a = 6 \), \( b = 13 \), and \( c = 5 \).

1. **Calculate the discriminant**:

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

2. **Take the square root of the discriminant**:

\[
\sqrt{49} = 7
\]

3. **Substitute \( a \), \( b \), and the square root of the discriminant into the formula**:

\[
x = \frac{{-13 \pm 7}}{{2 \cdot 6}} = \frac{{-13 \pm 7}}{{12}}
\]

4. **Calculate the two possible solutions**:

- **First solution**:

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

- **Second solution**:

\[
x_2 = \frac{{-13 - 7}}{{12}} = \frac{{-20}}{{12}} = -\frac{5}{3}
\]

Thus, the zeros of the quadratic equation \( y = 6x^2 + 13x + 5 \) are:

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