Question

Match the equivalent equations with the options provided:
y = (x + 7) (x + 2)
y = x^2 - 8x + 12
y = (x + 2)^2 - 5
y = x^2 + 10x + 19
Options:
y = x^2 + 4x - 1
y = x^2 + 9x + 14
y = (x + 5)^2 - 6
y = (x - 6) (x - 2)

Answer 1

To match the equivalent equations, we can simplify or expand each of the provided equations and see which ones are equivalent.

1. **First Equation:** \( y = (x + 7) (x + 2) \)

Expanding:
\[
y = x^2 + 2x + 7x + 14 = x^2 + 9x + 14
\]

2. **Second Equation:** \( y = x^2 - 8x + 12 \)

This equation is already in standard form and does not directly correspond to any in the options.

3. **Third Equation:** \( y = (x + 2)^2 - 5 \)

Expanding:
\[
y = (x^2 + 4x + 4) - 5 = x^2 + 4x - 1
\]

4. **Fourth Equation:** \( y = x^2 + 10x + 19 \)

This equation is also already in standard form and does not directly correspond to any in the options.

Now, we will match the equations with the options provided:

- From the first expansion:
- \( y = (x + 7)(x + 2) \) matches **\( y = x^2 + 9x + 14 \)**.

- From the third expansion:
- \( y = (x + 2)^2 - 5 \) matches **\( y = x^2 + 4x - 1 \)**.

The remaining equations:
- \( y = x^2 - 8x + 12 \) and \( y = x^2 + 10x + 19 \) do not correspond with the options given since each combination has been matched with an option.

Therefore, the matches are:

- \( y = (x + 7)(x + 2) \) ↔ **\( y = x^2 + 9x + 14 \)**
- \( y = (x + 2)^2 - 5 \) ↔ **\( y = x^2 + 4x - 1 \)**