Question

Match the equations to their equivalent in the options:
y = x^2 + 2x - 8
y = x^2 - 10x + 16
y = (x + 9) (x + 1)
y = (x + 3)^2 - 8
Options:
y = x^2 + 6x + 1
y = x^2 + 10x + 9
y = (x + 1)^2 - 9 (the only ^ was the 2, the -9 was not in the ^ as well)
y = (x - 2) (x - 8)

Answer 1

To match the given equations to their equivalents in the options, we need to analyze and simplify them step by step.

1. **Equation:** \(y = x^2 + 2x - 8\)
- This can be factored or completed to see if it matches any options.
- Completing the square:
\[
y = (x^2 + 2x + 1) - 1 - 8 = (x + 1)^2 - 9
\]
- **Match:** \(y = (x + 1)^2 - 9\)

2. **Equation:** \(y = x^2 - 10x + 16\)
- Completing the square:
\[
y = (x^2 - 10x + 25) - 25 + 16 = (x - 5)^2 - 9
\]
- This does not match any in the options directly, but we can check the expanded form:
\[
y = (x - 2)(x - 8)
\]
- **Match:** \(y = (x - 2)(x - 8)\)

3. **Equation:** \(y = (x + 9)(x + 1)\)
- Expanding:
\[
y = x^2 + 9x + 1x + 9 = x^2 + 10x + 9
\]
- **Match:** \(y = x^2 + 10x + 9\)

4. **Equation:** \(y = (x + 3)^2 - 8\)
- Expanding:
\[
y = (x^2 + 6x + 9) - 8 = x^2 + 6x + 1
\]
- **Match:** \(y = x^2 + 6x + 1\)

**Summary of matches:**
- \(y = x^2 + 2x - 8\) matches \(y = (x + 1)^2 - 9\)
- \(y = x^2 - 10x + 16\) matches \(y = (x - 2)(x - 8)\)
- \(y = (x + 9)(x + 1)\) matches \(y = x^2 + 10x + 9\)
- \(y = (x + 3)^2 - 8\) matches \(y = x^2 + 6x + 1\)