Question

odels. fill in the blanks:

1. $x^2 + 8x + 15 = (x + \underline{quad})(x + \underline{quad})$
2. $x^2 - 7x + 12 = (x - \underline{quad})(x - \underline{quad})$

Explanation:

Problem 2:

Step1: Find two numbers that multiply to 15 and add to 8.

We need two numbers \( a \) and \( b \) such that \( a \times b = 15 \) and \( a + b = 8 \). The factors of 15 are 1 & 15, 3 & 5. Among these, 3 and 5 add up to 8.

Step2: Factor the quadratic.

Using the numbers 3 and 5, we can factor \( x^2 + 8x + 15 \) as \( (x + 3)(x + 5) \).

Step1: Find two numbers that multiply to 12 and add to 7.

We need two numbers \( a \) and \( b \) such that \( a \times b = 12 \) and \( a + b = 7 \). The factors of 12 are 1 & 12, 2 & 6, 3 & 4. Among these, 3 and 4 add up to 7.

Step2: Factor the quadratic.

Since the middle term is \(-7x\) and the constant term is \(+12\), we use the numbers 3 and 4 with negative signs. So we can factor \( x^2 - 7x + 12 \) as \( (x - 3)(x - 4) \).

Answer:

The blanks are 3 and 5, so \( x^2 + 8x + 15 = (x + \boldsymbol{3})(x + \boldsymbol{5}) \)

Problem 4: