To convert the quadratic equation \( y = x^2 - 8x + 24 \) into vertex form, we can complete the square.
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Start with the original equation: \[ y = x^2 - 8x + 24 \]
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Focus on the \( x^2 - 8x \) part. To complete the square, calculate half of the coefficient of \( x \) (which is \(-8\)), and then square it: \[ \text{Half of } -8 = -4 \quad \Rightarrow \quad \text{Square it: } (-4)^2 = 16 \]
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Rewrite the equation, adding and subtracting 16 inside the equation: \[ y = (x^2 - 8x + 16) + 24 - 16 \] This simplifies to: \[ y = (x - 4)^2 + 8 \]
Now, the quadratic equation is in vertex form: \[ y = (x - 4)^2 + 8 \]
The vertex form is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. From our equation, we can identify:
- \( h = 4 \)
- \( k = 8 \)
Therefore, the coordinates of the vertex are: \[ (4, 8) \]
In summary, the vertex form of the quadratic is: \[ y = (x - 4)^2 + 8 \] And the coordinates of the vertex are \((4, 8)\).