https://file.aiquickdraw.com/mathbot/file/1725941889948dahevnwz.jpg

The given equation from the image is:

\[
(3x + 2)(2x + 3) = 6
\]

Let's solve this step-by-step:

### Step 1: Expand the left-hand side
Use the distributive property (FOIL method) to expand \((3x + 2)(2x + 3)\):

\[
(3x + 2)(2x + 3) = 3x \cdot 2x + 3x \cdot 3 + 2 \cdot 2x + 2 \cdot 3
\]

Simplifying each term:

\[
= 6x^2 + 9x + 4x + 6
\]

Now combine like terms:

\[
= 6x^2 + 13x + 6
\]

### Step 2: Write the complete equation
Now substitute this expanded form into the original equation:

\[
6x^2 + 13x + 6 = 6
\]

### Step 3: Move all terms to one side
Subtract 6 from both sides:

\[
6x^2 + 13x + 6 - 6 = 0
\]

This simplifies to:

\[
6x^2 + 13x = 0
\]

### Step 4: Factor the equation
Factor out \(x\) from both terms:

\[
x(6x + 13) = 0
\]

### Step 5: Solve for \(x\)
Now solve for \(x\) by setting each factor equal to zero:

1. \(x = 0\)
2. \(6x + 13 = 0\)

For the second equation:

\[
6x = -13 \quad \Rightarrow \quad x = -\frac{13}{6}
\]

### Final Solutions:
The solutions to the equation are:

\[
x = 0 \quad \text{or} \quad x = -\frac{13}{6}
\]

Would you like more details on any specific part? Here are 5 related questions:

1. What is the FOIL method in expanding binomials?
2. How do we solve quadratic equations by factoring?
3. Can we solve the quadratic equation using the quadratic formula instead of factoring?
4. How do we handle equations that have no real solutions?
5. What are common mistakes to avoid when expanding binomials?

**Tip:** Always double-check your expansion and factorization when solving equations involving polynomials to avoid mistakes.

Explore the solution to the quadratic equation (3x + 2)(2x + 3) = 6 with a complete walkthrough including expansion, simplification, and factorization steps. This guide is suitable for high school students learning advanced algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation