To solve the quadratic equation \( x^2 + 3x - 5 = 0 \), we'll use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 3 \), and \( c = -5 \).
Step 1: Calculate the Discriminant
First, we need to calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]
Substituting in our values for \( a \), \( b \), and \( c \):
\[ D = 3^2 - 4 \cdot 1 \cdot (-5) \]
Calculating \( D \):
\[ D = 9 + 20 = 29 \]
Step 2: Apply the Quadratic Formula
Now that we have the discriminant, we substitute \( a \), \( b \), and \( D \) back into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{29}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-3 \pm \sqrt{29}}{2} \]
Step 3: Write the Solutions
Thus, we can express the two solutions as:
\[ x_1 = \frac{-3 + \sqrt{29}}{2} \] \[ x_2 = \frac{-3 - \sqrt{29}}{2} \]
Conclusion
So the solutions to the quadratic equation \( x^2 + 3x - 5 = 0 \) are:
\[ x = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{2} \]
3 answers