To solve the problem, we need to find a path from the "Start" to the "Finish" by connecting expressions that are equivalent. This involves simplifying or expanding each expression and matching them with their equivalents.
Step-by-Step Solution:
1.
Start: The starting point is \( 2b(3b + 2) \).
- Simplify: \( 2b(3b + 2) = 6b^2 + 4b \).
2.
Next Step: From \( 6b^2 + 4b \), we need to find an equivalent expression.
- The expression \( 6b^2 + 4b \) matches with \( 2x(3x - 4) \) if we substitute \( x = b \).
3.
Next Step: From \( 2x(3x - 4) \), simplify:
- \( 2x(3x - 4) = 6x^2 - 8x \).
4.
Next Step: From \( 6x^2 - 8x \), we need to find an equivalent expression.
- The expression \( 6x^2 - 8x \) matches with \( -x(3x - 2) \) if we factor out \(-x\):
\[
6x^2 - 8x = -x(-6x + 8) = -x(3x - 2).
\]
5.
Next Step: From \( -x(3x - 2) \), simplify:
- \( -x(3x - 2) = -3x^2 + 2x \).
6.
Next Step: From \( -3x^2 + 2x \), we need to find an equivalent expression.
- The expression \( -3x^2 + 2x \) matches with \( 2x(4x + 2) \) if we factor out \( 2x \):
\[
-3x^2 + 2x = 2x(-\frac{3}{2}x + 1).
\]
- However, a simpler match is \( 2x(4x + 2) \) which simplifies to \( 8x^2 + 4x \).
7.
Next Step: From \( 2x(4x + 2) \), simplify:
- \( 2x(4x + 2) = 8x^2 + 4x \).
8.
Next Step: From \( 8x^2 + 4x \), we need to find an equivalent expression.
- The expression \( 8x^2 + 4x \) matches with \( 4x^2 - 4x \) if we factor out \( 4x \):
\[
8x^2 + 4x = 4x(2x + 1).
\]
- However, a simpler match is \( 4x^2 - 4x \) which simplifies to \( 4x(x - 1) \).
9.
Next Step: From \( 4x^2 - 4x \), we need to find an equivalent expression.
- The expression \( 4x^2 - 4x \) matches with \( 9x - 6 \) if we factor out \( 3 \):
\[
4x^2 - 4x = 4x(x - 1).
\]
- However, a simpler match is \( 9x - 6 \) which simplifies to \( 3(3x - 2) \).
10.
Next Step: From \( 9x - 6 \), simplify:
- \( 9x - 6 = 3(3x - 2) \).
11.
Finish: The final expression \( 3(3x - 2) \) leads directly to the "Finish".
Path Summary:
- Start: \( 2b(3b + 2) \)
- \( 6b^2 + 4b \)
- \( 2x(3x - 4) \)
- \( 6x^2 - 8x \)
- \( -x(3x - 2) \)
- \( -3x^2 + 2x \)
- \( 2x(4x + 2) \)
- \( 8x^2 + 4x \)
- \( 4x^2 - 4x \)
- \( 9x - 6 \)
- Finish: \( 3(3x - 2) \)
Final Answer:
\[
\boxed{2b(3b + 2) \rightarrow 6b^2 + 4b \rightarrow 2x(3x - 4) \rightarrow 6x^2 - 8x \rightarrow -x(3x - 2) \rightarrow -3x^2 + 2x \rightarrow 2x(4x + 2) \rightarrow 8x^2 + 4x \rightarrow 4x^2 - 4x \rightarrow 9x - 6 \rightarrow 3(3x - 2)}
\]
Parent Tip: Review the logic above to help your child master the concept of multiplying monomials worksheet pdf.
