Divide and simplify.

StartFraction 9 b plus 36 Over 4 a plus 12 EndFraction divided by StartFraction ab minus 3 b plus 4 a minus 12 Over a squared minus 6 a plus 9 EndFraction9b+364a+12÷ab−3b+4a−12a2−6a+9

Question content area bottom

Part 1

StartFraction 9 b plus 36 Over 4 a plus 12 EndFraction divided by StartFraction ab minus 3 b plus 4 a minus 12 Over a squared minus 6 a plus 9 EndFraction9b+364a+12÷ab−3b+4a−12a2−6a+9equals=enter your response here

Let's simplify this expression step by step.

### Expression:
\[
\frac{9b + 36}{4a + 12} \div \frac{ab - 3b + 4a - 12}{a^2 - 6a + 9}
\]

### Step 1: Factor both numerators and denominators
#### First Fraction:
\[
\frac{9b + 36}{4a + 12}
\]
- **Numerator**: \( 9b + 36 = 9(b + 4) \)
- **Denominator**: \( 4a + 12 = 4(a + 3) \)

So the first fraction becomes:
\[
\frac{9(b + 4)}{4(a + 3)}
\]

#### Second Fraction:
\[
\frac{ab - 3b + 4a - 12}{a^2 - 6a + 9}
\]
Group terms and factor:
- **Numerator**:
  \[
  ab - 3b + 4a - 12 = b(a - 3) + 4(a - 3) = (b + 4)(a - 3)
  \]
- **Denominator**:
  \[
  a^2 - 6a + 9 = (a - 3)^2
  \]

So the second fraction becomes:
\[
\frac{(b + 4)(a - 3)}{(a - 3)^2}
\]

### Step 2: Rewrite the division as multiplication by the reciprocal:
\[
\frac{9(b + 4)}{4(a + 3)} \div \frac{(b + 4)(a - 3)}{(a - 3)^2}
\]
This is equivalent to:
\[
\frac{9(b + 4)}{4(a + 3)} \times \frac{(a - 3)^2}{(b + 4)(a - 3)}
\]

### Step 3: Cancel common factors:
- \( (b + 4) \) appears in both the numerator and denominator, so cancel it.
- One factor of \( (a - 3) \) cancels out, leaving \( (a - 3) \) in the numerator.

Now the expression is:
\[
\frac{9(a - 3)}{4(a + 3)}
\]

### Final Answer:
\[
\frac{9(a - 3)}{4(a + 3)}
\]

Would you like a more detailed explanation of any step or have any further questions?

### Related Questions:
1. How do you factor quadratic expressions?
2. What is the process for simplifying algebraic fractions?
3. How do you apply the division of fractions rule?
4. What are the steps to cancel out common factors in algebraic expressions?
5. How do you identify common factors when simplifying complex fractions?

### Tip:
Always look for common factors to cancel before performing any operations when dealing with fractions!

Divide and simplify: \( \frac{9b + 36}{4a + 12} \div \frac{ab - 3b + 4a - 12}{a^2 - 6a + 9} \)

Learn how to divide and simplify the rational expression \( \frac{9b + 36}{4a + 12} \div \frac{ab - 3b + 4a - 12}{a^2 - 6a + 9} \) by factoring and canceling common factors. This solution involves algebraic concepts, including rational expressions and factoring, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation