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 our equation, the coefficients are:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -5 \)
Step 1: Calculate the discriminant
First, we need to calculate the discriminant, \( D \), which is given by \( D = b^2 - 4ac \).
\[ D = (3)^2 - 4(1)(-5) \] \[ D = 9 + 20 \] \[ D = 29 \]
Step 2: Apply the quadratic formula
Now that we have the discriminant, we can find the values of \( x \).
\[ x = \frac{-3 \pm \sqrt{29}}{2(1)} \] \[ x = \frac{-3 \pm \sqrt{29}}{2} \]
Step 3: Write the final solutions
Thus, the solutions are:
\[ x_1 = \frac{-3 + \sqrt{29}}{2} \] \[ x_2 = \frac{-3 - \sqrt{29}}{2} \]
These represent the two solutions to the quadratic equation \( x^2 + 3x - 5 = 0 \).
Summary of the solutions:
\[ x_1 = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x_2 = \frac{-3 - \sqrt{29}}{2} \]