https://file.aiquickdraw.com/mathbot/file/17259625988632ay0aquc.jpg

The equation presented is:

\[
(ax + 1)(x - b) = cx - x^2 - 2
\]

We are asked to find the value of \(c\), given that this equation holds for all \(x \in \mathbb{R}\).

### Step 1: Expand the left-hand side
Let's expand the left-hand side of the equation:

\[
(ax + 1)(x - b) = ax(x - b) + 1(x - b)
\]
\[
= ax^2 - abx + x - b
\]

Thus, the expanded form is:

\[
ax^2 - abx + x - b
\]

### Step 2: Rearrange both sides
We now rewrite the expanded equation:

\[
ax^2 - abx + x - b = cx - x^2 - 2
\]

### Step 3: Compare coefficients
For this equation to hold for all \(x\), we need to compare the coefficients of the powers of \(x\) on both sides. First, let's group terms by powers of \(x\):

1. The coefficient of \(x^2\): 
   - On the left: \(a\)
   - On the right: \(-1\)
   
   Therefore, \(a = -1\).

2. The coefficient of \(x\):
   - On the left: \(-ab + 1\)
   - On the right: \(c\)
   
   Therefore, \(-ab + 1 = c\).

3. The constant term:
   - On the left: \(-b\)
   - On the right: \(-2\)
   
   Therefore, \(-b = -2\), so \(b = 2\).

### Step 4: Solve for \(c\)
Now that we know \(b = 2\) and \(a = -1\), we can substitute these values into the equation \(-ab + 1 = c\):

\[
-(-1)(2) + 1 = c
\]
\[
2 + 1 = c
\]
\[
c = 3
\]

Thus, the value of \(c\) is **3**, so the correct answer is **E**.

---

Do you want more details or have any questions? 

Here are 5 related questions to explore:
1. How can the same technique of comparing coefficients be applied to other polynomial equations?
2. What are the conditions under which a quadratic equation has real solutions?
3. How do you factor polynomials to solve equations like this one?
4. What is the significance of solutions holding for all \(x \in \mathbb{R}\)?
5. How does the degree of a polynomial affect the number of solutions it has?

**Tip:** When solving equations with unknown coefficients, expanding and comparing coefficients of like terms is a powerful method to isolate variables.

Persamaan (ax + 1)(x - b) = cx - x^2 - 2 berlaku untuk semua x ∈ R, maka nilai c = ....

Learn how to solve the quadratic equation (ax + 1)(x - b) = cx - x^2 - 2 by expanding polynomials and comparing coefficients. Understand how to find the unknown value of c. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation