To find which given \( x \) value produces the largest output for the function defined by the equation:
\[ \frac{1}{2}x = -x^2 + 4, \]
we can first rewrite it into a more manageable form. Multiplying both sides by 2 to eliminate the fraction gives us:
\[ x = -2x^2 + 8. \]
This simplifies to:
\[ 2x^2 + x - 8 = 0. \]
Now, we can solve this quadratic equation to find the \( x \) values. We will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \( a = 2, b = 1, c = -8 \).
Calculating the discriminant \( b^2 - 4ac \):
\[ 1^2 - 4(2)(-8) = 1 + 64 = 65. \]
Now we can plug in the values into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{65}}{2(2)} = \frac{-1 \pm \sqrt{65}}{4}. \]
This will yield two \( x \) values \( x_1 \) and \( x_2 \).
Next, we need to evaluate the function at the given \( x \) values: -1, 1, 0, and 2.
Substituting these \( x \) values into the original transformed equation \( y = -x^2 + 4 \) to find \( y \) values:
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For \( x = -1 \): \[ y = -(-1)^2 + 4 = -1 + 4 = 3. \]
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For \( x = 1 \): \[ y = -(1)^2 + 4 = -1 + 4 = 3. \]
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For \( x = 0 \): \[ y = -(0)^2 + 4 = 0 + 4 = 4. \]
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For \( x = 2 \): \[ y = -(2)^2 + 4 = -4 + 4 = 0. \]
Now we have the outputs for each value:
- For \( x = -1 \): \( y = 3 \)
- For \( x = 1 \): \( y = 3 \)
- For \( x = 0 \): \( y = 4 \)
- For \( x = 2 \): \( y = 0 \)
The largest output occurs at \( x = 0 \) with a value of \( 4 \).
Thus, the given \( X \) value that produces the largest output is \( \boxed{0} \).