To solve the equation where half of \( x \) equals the expression \(-12x^2 + 21x - 3\), we can rewrite the equation as:
\[ \frac{1}{2} x = -12x^2 + 21x - 3 \]
To eliminate the fraction, we can multiply both sides by 2:
\[ x = -24x^2 + 42x - 6 \]
Now we want to rearrange this equation into a standard quadratic form by bringing all terms to one side:
\[ 24x^2 - 41x + 6 = 0 \]
Next, we can use the quadratic formula to solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 24 \), \( b = -41 \), and \( c = 6 \). Plugging these values into the formula gives us:
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Calculate the discriminant: \[ b^2 - 4ac = (-41)^2 - 4 \cdot 24 \cdot 6 = 1681 - 576 = 1105 \]
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Now plug the values into the quadratic formula: \[ x = \frac{41 \pm \sqrt{1105}}{48} \]
The solutions for \( x \) would be:
\[ x_1 = \frac{41 + \sqrt{1105}}{48} \] \[ x_2 = \frac{41 - \sqrt{1105}}{48} \]
Next, the problem references two tables: Table 1 has values of \( x \) (0, 1, 2) and their corresponding \( f(x) \) being -369. We'll explore the function \( f(x) \):
- \( f(0) = -369 \)
- \( f(1) = -369 \)
- \( f(2) = -369 \)
Since all outputs of \( f(x) \) equal -369, it seems \( f(x) \) is constant.
For Table 2 where \( g(x) \) is defined similarly, we may also evaluate the function \( g(x) \) at the same \( x \) values of 0, 1, and 2. Since \( g(x) \) values are not provided, we can't provide specific outputs unless more information for \( g(x) \) is supplied.
If you provide a specific form or data for \( g(x) \), I can help evaluate further. However, if this is simply a structure for two pieces of data, we can summarize that \( f(x) \) is constant at -369 for defined \( x \) values, while \( g(x) \) remains undefined or requires additional information.