Solve the equation `x^(2)-45x+324=0`.

Answer

To solve the quadratic equation \(x^2 - 45x + 324 = 0\), we will use the middle term splitting method. Here are the steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 1\)
- \(b = -45\)
- \(c = 324\)

### Step 2: Calculate the product \(ac\)
We need to calculate the product \(ac\):
\[ ac = 1 \times 324 = 324 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(324\) (the value of \(ac\)) and add up to \(-45\) (the value of \(b\)).

We can list the factor pairs of \(324\):
- \(1 \times 324\)
- \(2 \times 162\)
- \(3 \times 108\)
- \(4 \times 81\)
- \(6 \times 54\)
- \(9 \times 36\)
- \(12 \times 27\)
- \(18 \times 18\)

Now, we need to find a pair that adds up to \(-45\). Since the sum is negative, both numbers must be negative:
- \(-9\) and \(-36\) satisfy this condition because:
\[ -9 + (-36) = -45 \quad \text{and} \quad (-9) \times (-36) = 324 \]

### Step 4: Rewrite the equation using the found numbers
We can now rewrite the middle term \(-45x\) as \(-9x - 36x\):
\[ x^2 - 9x - 36x + 324 = 0 \]

### Step 5: Factor the quadratic expression
Now, we group the terms:
\[ (x^2 - 9x) + (-36x + 324) = 0 \]
Factoring out common terms:
\[ x(x - 9) - 36(x - 9) = 0 \]
Now we can factor by grouping:
\[ (x - 9)(x - 36) = 0 \]

### Step 6: Set each factor to zero
Now, we set each factor to zero:
1. \(x - 9 = 0 \implies x = 9\)
2. \(x - 36 = 0 \implies x = 36\)

### Conclusion
The solutions to the equation \(x^2 - 45x + 324 = 0\) are:
\[ x = 9 \quad \text{and} \quad x = 36 \]

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