Determine the zero(s) of the quadratic function y=6x^2+13x+5. Use Algebra (factoring or quadratic formula) to prove your answer. Please write it simply.

Question

GPT-4o mini · Answer

To find the zeros of the quadratic function $y = 6 x^{2} + 13 x + 5$ , we can use the quadratic formula, which is:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

In our quadratic function, the coefficients are:

$a = 6$
$b = 13$
$c = 5$

Now, we first need to calculate the discriminant $b^{2} - 4 a c$ :

$b^{2} = 13^{2} = 169$ $4 a c = 4 \times 6 \times 5 = 120$ $b^{2} - 4 a c = 169 - 120 = 49$

Next, we compute the zeros using the quadratic formula:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 13 \pm \sqrt{49}}{2 \times 6}$ $x = \frac{- 13 \pm 7}{12}$

Now we can calculate the two possible values for $x$ :

Using $- 13 + 7$ : $x = \frac{- 6}{12} = - \frac{1}{2}$
Using $- 13 - 7$ : $x = \frac{- 20}{12} = - \frac{5}{3}$

The zeros of the function $y = 6 x^{2} + 13 x + 5$ are:

$x = - \frac{1}{2} and x = - \frac{5}{3}$

Thus, the final answer is:

$- \frac{1}{2}, - \frac{5}{3}$