Question

Solve the quadratic equation. Show all of your steps.

x2+3x−5=0

Answer 1

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:

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