Solve `x^2 -38x+280=0`

Answer

To solve the quadratic equation \(x^2 - 38x + 280 = 0\), we can use the method of middle term factorization. Here are the steps to solve the equation:

### Step 1: Write the equation
We start with the equation:
\[ x^2 - 38x + 280 = 0 \]

### Step 2: Identify the coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), we have:
- \(a = 1\)
- \(b = -38\)
- \(c = 280\)

### Step 3: Find two numbers that multiply to \(c\) and add to \(b\)
We need to find two numbers that multiply to \(280\) (the constant term) and add to \(-38\) (the coefficient of \(x\)).

After testing several pairs, we find:
- The numbers \(-28\) and \(-10\) satisfy this because:
- \(-28 \times -10 = 280\)
- \(-28 + -10 = -38\)

### Step 4: Rewrite the middle term
We can rewrite the equation using these two numbers:
\[ x^2 - 28x - 10x + 280 = 0 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (x^2 - 28x) + (-10x + 280) = 0 \]
Factoring out the common terms from each group, we get:
\[ x(x - 28) - 10(x - 28) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((x - 28)\):
\[ (x - 28)(x - 10) = 0 \]

### Step 7: Set each factor to zero
Now, we set each factor equal to zero:
1. \(x - 28 = 0\) → \(x = 28\)
2. \(x - 10 = 0\) → \(x = 10\)

### Step 8: Write the final solution
The solutions to the equation \(x^2 - 38x + 280 = 0\) are:
\[ x = 28 \quad \text{and} \quad x = 10 \]