Question

function g is defined by g(x)=3x^2-2x-5. what are the solutions to g(x)=0?

Responses:
x=-1 and x=3/5
x=-1 and x=5/3
x=1 and x=-3/5
x=1 and x=-5/3

Answer 1

To find the solutions to the equation \( g(x) = 0 \) for the function \( g(x) = 3x^2 - 2x - 5 \), we need to solve the quadratic equation \( 3x^2 - 2x - 5 = 0 \).

We can use the quadratic formula:

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

where \( a = 3 \), \( b = -2 \), and \( c = -5 \).

1. Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-2)^2 - 4(3)(-5) = 4 + 60 = 64
\]

2. Now plug the values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{64}}{2(3)} = \frac{2 \pm 8}{6}
\]

This leads to two possible solutions:

- For the positive case:
\[
x = \frac{2 + 8}{6} = \frac{10}{6} = \frac{5}{3}
\]

- For the negative case:
\[
x = \frac{2 - 8}{6} = \frac{-6}{6} = -1
\]

Thus, the solutions to \( g(x) = 0 \) are \( x = -1 \) and \( x = \frac{5}{3} \).

So the correct response is:
**x = -1 and x = 5/3**.