To solve the quadratic equation \( x^2 + 3x - 5 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a \), \( b \), and \( c \) are coefficients from the general form of a quadratic equation \( ax^2 + bx + c = 0 \). For our equation:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -5 \)
Now we can substitute these values into the quadratic formula.
Step 1: Calculate the discriminant
First, we need to calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-5) \] \[ = 9 + 20 \] \[ = 29 \]
Step 2: Substitute into the quadratic formula
Now, substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{29}}{2 \cdot 1} \] \[ = \frac{-3 \pm \sqrt{29}}{2} \]
Step 3: Write the solutions
The solutions to the equation are:
\[ x = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{2} \]
Final Answer
Thus, the solutions to the quadratic equation \( x^2 + 3x - 5 = 0 \) are:
\[ x = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{2} \]