Solve the following quadratic equations by completing the squares.
`x^(2) + 3x - 5 = 0`

Answer

To solve the quadratic equation \( x^2 + 3x - 5 = 0 \) by completing the square, follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ x^2 + 3x - 5 = 0 \]
Add 5 to both sides to isolate the quadratic and linear terms:
\[ x^2 + 3x = 5 \]

### Step 2: Complete the square
To complete the square, we need to add and subtract the square of half the coefficient of \( x \). The coefficient of \( x \) is 3, so half of this is \( \frac{3}{2} \). Now, square this value:
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
Add and subtract \( \frac{9}{4} \) to the left side:
\[ x^2 + 3x + \frac{9}{4} - \frac{9}{4} = 5 \]
This simplifies to:
\[ \left(x + \frac{3}{2}\right)^2 - \frac{9}{4} = 5 \]

### Step 3: Move the constant to the other side
Now, add \( \frac{9}{4} \) to both sides:
\[ \left(x + \frac{3}{2}\right)^2 = 5 + \frac{9}{4} \]
To add these, convert 5 into a fraction with a denominator of 4:
\[ 5 = \frac{20}{4} \]
So,
\[ 5 + \frac{9}{4} = \frac{20}{4} + \frac{9}{4} = \frac{29}{4} \]
Thus, we have:
\[ \left(x + \frac{3}{2}\right)^2 = \frac{29}{4} \]

### Step 4: Take the square root of both sides
Taking the square root of both sides gives:
\[ x + \frac{3}{2} = \pm \sqrt{\frac{29}{4}} \]
This simplifies to:
\[ x + \frac{3}{2} = \pm \frac{\sqrt{29}}{2} \]

### Step 5: Solve for \( x \)
Now, isolate \( x \):
\[ x = -\frac{3}{2} \pm \frac{\sqrt{29}}{2} \]
Combining the fractions gives:
\[ x = \frac{-3 \pm \sqrt{29}}{2} \]

### Final Solution
The solutions to the equation \( x^2 + 3x - 5 = 0 \) are:
\[ x = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{2} \]

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