Question

Function g is defined by g(x) = 3x ^ 2 - 2x - 5 What are the solutions to g(x) = 0
x=4\& x = 5
x= 4 3 \& x = 5/3
x=-1\& x = 5/3

Answer 1

To find the solutions to \( g(x) = 0 \) 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:

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

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

First, we calculate the discriminant (\( D \)):

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

Now we plug \( D \) back into the quadratic formula:

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

This gives us two possible solutions:

1. \( x = \frac{2 + 8}{6} = \frac{10}{6} = \frac{5}{3} \)
2. \( x = \frac{2 - 8}{6} = \frac{-6}{6} = -1 \)

Thus, the solutions to the equation \( g(x) = 0 \) are:

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

Therefore, the correct answer is:

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