Solve `x^(2) - 10x + 21= 0`

Answer

To solve the quadratic equation \( x^2 - 10x + 21 = 0 \), we can use the method of middle-term factorization. Here are the steps:

### Step 1: Write down the equation
We start with the equation:
\[ x^2 - 10x + 21 = 0 \]

### Step 2: Factor the quadratic expression
We need to factor the quadratic expression \( x^2 - 10x + 21 \). We look for two numbers that multiply to \( 21 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( -3 \) and \( -7 \) because:
\[ -3 \times -7 = 21 \quad \text{and} \quad -3 + -7 = -10 \]

So, we can rewrite the quadratic as:
\[ x^2 - 3x - 7x + 21 = 0 \]

### Step 3: Group the terms
Now, we group the terms:
\[ (x^2 - 3x) + (-7x + 21) = 0 \]

### Step 4: Factor by grouping
Next, we factor out the common terms in each group:
\[ x(x - 3) - 7(x - 3) = 0 \]

Now, we can factor out \( (x - 3) \):
\[ (x - 3)(x - 7) = 0 \]

### Step 5: Set each factor to zero
Now, we set each factor equal to zero:
1. \( x - 3 = 0 \)
2. \( x - 7 = 0 \)

### Step 6: Solve for \( x \)
Solving these equations gives us:
1. \( x = 3 \)
2. \( x = 7 \)

### Final Answer
Thus, the solutions to the equation \( x^2 - 10x + 21 = 0 \) are:
\[ x = 3 \quad \text{and} \quad x = 7 \]

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