To match the given equations with their equivalents, we'll first rewrite and simplify each of the provided equations and the equivalents where necessary.
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For \( Y = x^2 - 10x + 16 \):
- This can be factored as follows: \[ Y = (x - 2)(x - 8) \] So, it matches with \( y = (x-2)(x-8) \).
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For \( y = (x + 3)^2 - 8 \):
- Expanding the square: \[ y = (x^2 + 6x + 9) - 8 = x^2 + 6x + 1 \] So, it matches with \( y = x^2 + 6x + 1 \).
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For \( y = (x + 9)(x + 1) \):
- Expanding this: \[ y = x^2 + 10x + 9 \] So, it matches with \( y = x^2 + 10x + 9 \).
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For \( y = x^2 + 2x - 8 \):
- This can be rewritten and completed to find the equivalent in vertex form: \[ y = (x + 1)^2 - 9 \] So, it matches with \( y = (x + 1)^2 - 9 \).
Thus, the matches are:
- \( Y = x^2 - 10x + 16 \) ↔ \( y = (x-2)(x-8) \)
- \( y = (x + 3)^2 - 8 \) ↔ \( y = x^2 + 6x + 1 \)
- \( y = (x + 9)(x + 1) \) ↔ \( y = x^2 + 10x + 9 \)
- \( y = x^2 + 2x - 8 \) ↔ \( y = (x + 1)^2 - 9 \)