Solve the quadratic equation. Show all of your steps.

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 the coefficients from \( ax^2 + bx + c = 0 \). Here, we have:

• \( a = 1 \)
• \( b = 3 \)
• \( c = -5 \)

Step 1: Calculate the Discriminant

First, we need to calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-5) \]

Calculating \( 3^2 \):

\[ 3^2 = 9 \]

Now calculate \( -4 \cdot 1 \cdot (-5) \):

\[ -4 \cdot 1 \cdot (-5) = 20 \]

Now combine these values:

\[ b^2 - 4ac = 9 + 20 = 29 \]

Step 2: Apply the Quadratic Formula

Now we can substitute \( a \), \( b \), and the discriminant into the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{29}}{2 \cdot 1} \]

This simplifies to:

\[ x = \frac{-3 \pm \sqrt{29}}{2} \]

Final Steps: Write the Solutions

Thus, the solutions to the quadratic equation \( x^2 + 3x - 5 = 0 \) are:

\[ x = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{2} \]

These are the exact solutions. You can further evaluate them numerically if needed, but they represent the exact values of \( x \).