To solve the quadratic equation \( 5x^2 + x - 1 = 0 \) using the quadratic formula, we first identify the coefficients in the standard form \( ax^2 + bx + c = 0 \):
- \( a = 5 \)
- \( b = 1 \)
- \( c = -1 \)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 1^2 - 4(5)(-1) = 1 + 20 = 21 \]
Next, we substitute the values of \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{21}}{2 \cdot 5} = \frac{-1 \pm \sqrt{21}}{10} \]
Thus, the two solutions are:
\[ x = \frac{-1 + \sqrt{21}}{10} \quad \text{and} \quad x = \frac{-1 - \sqrt{21}}{10} \]
Now, comparing these solutions to the provided answer options:
- The correct expressions are:
\[ x = \frac{-1 + \sqrt{21}}{10}, \quad x = \frac{-1 - \sqrt{21}}{10} \]
So, the closest match to the solutions you provided is:
Answer: \( x = \frac{10}{-1 + \sqrt{21}}, \quad x = \frac{10}{-1 - \sqrt{21}} \)