Solve the equation `4x^2+9=0` by factorization method.

Explanation

To solve the equation \(4x^2 + 9 = 0\) using the factorization method, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4x^2 + 9 = 0 \]

### Step 2: Move the constant to the other side
Subtract 9 from both sides:
\[ 4x^2 = -9 \]

### Step 3: Divide by 4
Divide both sides by 4 to isolate \(x^2\):
\[ x^2 = -\frac{9}{4} \]

### Step 4: Introduce imaginary unit \(i\)
Since we have a negative value on the right side, we can express it using the imaginary unit \(i\), where \(i = \sqrt{-1}\):
\[ x^2 = \frac{9}{4} \cdot (-1) = \frac{9}{4} \cdot i^2 \]

### Step 5: Take the square root
Taking the square root of both sides gives:
\[ x = \pm \sqrt{\frac{9}{4} \cdot i^2} \]
This simplifies to:
\[ x = \pm \frac{3}{2}i \]

### Step 6: Write the final roots
Thus, the roots of the equation \(4x^2 + 9 = 0\) are:
\[ x = \frac{3}{2}i \quad \text{and} \quad x = -\frac{3}{2}i \]

### Summary of the solution:
The roots of the quadratic equation \(4x^2 + 9 = 0\) are \(x = \frac{3}{2}i\) and \(x = -\frac{3}{2}i\).

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