To complete the square for the quadratic function \( y = -3x^2 - 54x - 236 \), we will follow these steps:
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Factor out the coefficient of \( x^2 \) from the first two terms: \[ y = -3(x^2 + 18x) - 236 \]
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Complete the square inside the parentheses. To do this, take half of the coefficient of \( x \) (which is \( 18 \)), square it, and then add and subtract it inside the parentheses:
- Half of \( 18 \) is \( 9 \), and \( 9^2 = 81 \).
So, we rewrite the equation as: \[ y = -3(x^2 + 18x + 81 - 81) - 236 \]
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Rewrite it: \[ y = -3((x + 9)^2 - 81) - 236 \]
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Distribute the \(-3\): \[ y = -3(x + 9)^2 + 243 - 236 \]
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Combine the constant terms: \[ y = -3(x + 9)^2 + 7 \]
Now, we have rewritten the equation in vertex form. The vertex form of the quadratic function is: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex.
For this function, we have: \[ y = -3(x + 9)^2 + 7 \]
The vertex is \((-9, 7)\).