To match each equation to its equivalent, we will simplify and factor each one as needed.
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For \( y = x^2 - 6x + 16 \): We can complete the square or look for factorizations, but it seems simpler to check and arrive at its equivalent: \[ y = (x - 3)^2 + 7 \] This shows that the vertex form of the quadratic confirms it's equivalent.
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For \( y = (x + 8)(x + 11) \): Expanding this: \[ y = x^2 + 19x + 88 \] This matches with \( y = x^2 + 19x + 88 \).
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For \( y = x^2 - 1x - 20 \): To factor this, we look for two numbers that multiply to -20 and add to -1. We find: \[ y = (x + 4)(x - 5) \]
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For \( y = (x + 4)^2 - 5 \): Expanding gives: \[ y = x^2 + 8x + 16 - 5 = x^2 + 8x + 11 \]
In summary, the matches are:
- \( y = x^2 - 6x + 16 \) → \( y = (x - 3)^2 + 7 \)
- \( y = (x + 8)(x + 11) \) → \( y = x^2 + 19x + 88 \)
- \( y = x^2 - 1x - 20 \) → \( y = (x + 4)(x - 5) \)
- \( y = (x + 4)^2 - 5 \) → \( y = x^2 + 8x + 11 \)