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 \( c = -1 \).
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-1) = 25 + 8 = 33 \]
- 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 \( g(x) = 0 \).
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