Which of the following is the factored form of `x^2- 7x + 10` ?

Options

A. `(x-7)(x-3)`
B. `(x+4)(x+3)`
C. `(x-4)(x-3)`
D. `(x-5)(x-2)`

Answer

To factor the quadratic expression \(x^2 - 7x + 10\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = -7\) (coefficient of \(x\))
- \(c = 10\) (constant term)

### Step 2: Find two numbers that multiply to \(c\) and add to \(b\)
We need to find two numbers that multiply to \(c = 10\) and add up to \(b = -7\). The two numbers that satisfy these conditions are \(-5\) and \(-2\) because:
- \(-5 \times -2 = 10\)
- \(-5 + -2 = -7\)

### Step 3: Rewrite the middle term
We can rewrite the expression \(x^2 - 7x + 10\) using the two numbers found:
\[ x^2 - 5x - 2x + 10 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 - 5x) + (-2x + 10) \]

### Step 5: Factor by grouping
Now, we factor out the common factors from each group:
- From the first group \(x^2 - 5x\), we can factor out \(x\):
\[ x(x - 5) \]
- From the second group \(-2x + 10\), we can factor out \(-2\):
\[ -2(x - 5) \]

### Step 6: Combine the factors
Now we can combine the factored groups:
\[ x(x - 5) - 2(x - 5) \]
This can be factored further by taking out the common factor \((x - 5)\):
\[ (x - 5)(x - 2) \]

### Step 7: Write the final factored form
Thus, the factored form of the expression \(x^2 - 7x + 10\) is:
\[ (x - 5)(x - 2) \]

### Final Answer
The factored form of \(x^2 - 7x + 10\) is \((x - 5)(x - 2)\).

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