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The problem asks to express \( x^2 + 6x \) in the form \( (x + a)^2 + b \), where \( a \) and \( b \) are constants.

To solve this, we need to **complete the square** on \( x^2 + 6x \).

### Step 1: Half the coefficient of \( x \)
The coefficient of \( x \) is 6. Half of 6 is \( 3 \), so we can write:
\[
(x + 3)^2 = x^2 + 6x + 9
\]
Notice that we introduced a \( +9 \) term.

### Step 2: Subtract the extra term
Since we added \( 9 \), we need to subtract \( 9 \) to maintain equality:
\[
x^2 + 6x = (x + 3)^2 - 9
\]

### Conclusion
Thus, the expression \( x^2 + 6x \) can be written as:
\[
(x + 3)^2 - 9
\]
From this, we can identify:
\[
a = 3 \quad \text{and} \quad b = -9
\]

Would you like further details on this method?

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Here are 5 related questions for further practice:
1. How would you complete the square for \( x^2 + 8x \)?
2. Can you express \( x^2 - 4x \) in the form \( (x + a)^2 + b \)?
3. What are the values of \( a \) and \( b \) when completing the square for \( x^2 + 10x \)?
4. How does completing the square help in solving quadratic equations?
5. Can you derive the quadratic formula by completing the square?

**Tip:** Completing the square is essential for solving quadratic equations and understanding the geometry of parabolas!

Express x^2 + 6x in the form (x + a)^2 + b, where a and b are numbers. What are the values of a and b?

Learn how to express x^2 + 6x by completing the square to write it in the form (x + a)^2 + b. This problem introduces key algebraic concepts of transforming quadratic expressions and is suitable for Grades 9-10.

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