Match the equivalent equations with the options provided:

Question

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:

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