https://file.aiquickdraw.com/mathbot/file/1727111036485530ox4d8.jpg

The given expression is \( 4x^2 + bx - 45 \), and we need to rewrite it as \( (hx + k)(x + j) \), where \( h \), \( k \), and \( j \) are integer constants. The problem asks which of the following must be an integer, with options involving \( \frac{b}{h} \), \( \frac{b}{k} \), \( \frac{45}{h} \), and \( \frac{45}{k} \).

### Step 1: Expand the factorization
We start by expanding \( (hx + k)(x + j) \):
\[
(hx + k)(x + j) = hx^2 + hjx + kx + kj = hx^2 + (hj + k)x + kj
\]
This must match the form \( 4x^2 + bx - 45 \). So, comparing coefficients:
- Coefficient of \( x^2 \): \( h = 4 \)
- Coefficient of \( x \): \( hj + k = b \)
- Constant term: \( kj = -45 \)

### Step 2: Solve for \( k \) and \( j \)
Now we solve for integer pairs \( k \) and \( j \) such that \( kj = -45 \). The factor pairs of \( -45 \) are:
\[
(k, j) \in \{(-1, 45), (1, -45), (-3, 15), (3, -15), (-5, 9), (5, -9)\}
\]

### Step 3: Test for \( h = 4 \)
Substitute \( h = 4 \) into the equation \( hj + k = b \), i.e., \( 4j + k = b \). For each pair of \( k \) and \( j \), calculate \( b \).

- For \( (k, j) = (-1, 45) \): \( 4(45) - 1 = 179 \)
- For \( (k, j) = (1, -45) \): \( 4(-45) + 1 = -179 \)
- For \( (k, j) = (-3, 15) \): \( 4(15) - 3 = 57 \)
- For \( (k, j) = (3, -15) \): \( 4(-15) + 3 = -57 \)
- For \( (k, j) = (-5, 9) \): \( 4(9) - 5 = 31 \)
- For \( (k, j) = (5, -9) \): \( 4(-9) + 5 = -31 \)

Thus, possible values for \( b \) are \( 179, -179, 57, -57, 31, -31 \).

### Step 4: Determine the correct answer
Now, we check which of the given options must be an integer:
- \( \frac{b}{h} \): Since \( h = 4 \) and \( b \) takes values like 179, 57, etc., \( \frac{b}{4} \) is not necessarily an integer.
- \( \frac{b}{k} \): Since \( k \) can be 1, -1, 3, -3, etc., \( \frac{b}{k} \) is not always an integer.
- \( \frac{45}{h} \): This simplifies to \( \frac{45}{4} \), which is not an integer.
- \( \frac{45}{k} \): Since \( k \) is a divisor of \( -45 \), \( \frac{45}{k} \) will always be an integer.

Therefore, the correct answer is:
\[
\boxed{D} \, \frac{45}{k}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can we solve quadratic equations by factoring in general?
2. What is the relationship between the coefficients in a quadratic equation and its factorization?
3. How do we determine possible integer factor pairs for a given constant?
4. Can this method be extended to quadratic equations with leading coefficients other than 4?
5. What are other methods to factor quadratic expressions if simple factoring does not work?

**Tip:** When factoring quadratics, always look for factor pairs of the constant term that satisfy the middle term coefficient.

The expression 4x^2 + bx - 45, where b is a constant, can be rewritten as (hx + k)(x + j), where h, k, and j are integer constants. Which of the following must be an integer? (A) b/h, (B) b/k, (C) 45/h, (D) 45/k.

Explore how to factor the quadratic expression 4x^2 + bx - 45 and determine which of the given expressions must be an integer. This problem highlights key algebraic concepts including factorization and finding integer factor pairs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation