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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)

1 answer

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 \(y = x^2 + 6x + 1\)