Question

function g is defined as g(x) = 2x^2 + 5x - 1. what are the solutions to g(x) = 0

Answer 1

To find the solutions to the equation \( g(x) = 0 \) where \( g(x) = 2x^2 + 5x - 1 \), we need to solve the quadratic equation:

\[
2x^2 + 5x - 1 = 0
\]

We can use the quadratic formula:

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

where \( a = 2 \), \( b = 5 \), and \( c = -1 \).

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

\[
b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-1) = 25 + 8 = 33
\]

2. Now substitute \( a \), \( b \), and the discriminant into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{33}}{2 \cdot 2}
\]
\[
x = \frac{-5 \pm \sqrt{33}}{4}
\]

Thus, the two solutions are:

\[
x_1 = \frac{-5 + \sqrt{33}}{4}
\]
\[
x_2 = \frac{-5 - \sqrt{33}}{4}
\]

These are the solutions to the equation \( g(x) = 0 \).