Question:\(\mathrm{y = (x + 3)^2 - 4}\)y = -4Which ordered pair (x, y) is a solution to the system of...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = (x + 3)^2 - 4}\) (quadratic function)
  • \(\mathrm{y = -4}\) (horizontal line)
• Need to find: The ordered pair \(\mathrm{(x, y)}\) that satisfies both equations

2. INFER the solution strategy

• Since both equations are solved for y, I can set the right sides equal to each other
• This eliminates y and gives me an equation with only x
• Strategy: \(\mathrm{(x + 3)^2 - 4 = -4}\)

3. SIMPLIFY to solve for x

• Start with: \(\mathrm{(x + 3)^2 - 4 = -4}\)
• Add 4 to both sides: \(\mathrm{(x + 3)^2 = 0}\)
• Take the square root: \(\mathrm{x + 3 = 0}\)
• Solve for x: \(\mathrm{x = -3}\)

4. INFER the y-coordinate

• Since we need \(\mathrm{y = -4}\) (from the second equation), our solution is \(\mathrm{(-3, -4)}\)

5. Verify the solution

• Check in first equation: \(\mathrm{y = (-3 + 3)^2 - 4 = 0^2 - 4 = -4}\) ✓
• Check in second equation: \(\mathrm{y = -4}\) ✓

Answer: B. \(\mathrm{(-3, -4)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that both equations equal y, so the expressions can be set equal to each other. Instead, students might try to substitute answer choices one by one into both equations, which is time-consuming and doesn't demonstrate understanding of system-solving techniques. This approach can work but often leads to computational errors or confusion about which values to test first.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic mistakes when solving \(\mathrm{(x + 3)^2 - 4 = -4}\). Common errors include:

• Forgetting to add 4 to both sides, leaving \(\mathrm{(x + 3)^2 = -4}\) (which has no real solutions)
• Sign errors when taking the square root
• Incorrectly solving \(\mathrm{x + 3 = 0}\)

These algebraic missteps can lead students to select incorrect answer choices or become confused and guess.

The Bottom Line:

This problem tests whether students can efficiently solve systems by elimination (setting equal expressions equal) rather than just substitution checking. The key insight is recognizing that when two different expressions both equal the same variable, those expressions must equal each other.