To solve the problem, we'll start with the equation you've provided:
\[ \frac{1}{2} x = -12x^2 + 21x - 3 \]
First, we will rearrange the equation so that we can easily work with it:
\[ 0 = -12x^2 + 21x - 3 - \frac{1}{2}x \]
To eliminate the fraction, we can multiply the entire equation by 2:
\[ 0 = -24x^2 + 42x - 6 - x \] \[ 0 = -24x^2 + 41x - 6 \]
Next, we can rearrange the equation to set it to standard quadratic form:
\[ 24x^2 - 41x + 6 = 0 \]
To find the roots of this quadratic equation, we can use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 24\), \(b = -41\), and \(c = 6\).
- Calculate the discriminant, \(b^2 - 4ac\):
\[ (-41)^2 - 4(24)(6) = 1681 - 576 = 1105 \]
- Now substitute the values into the quadratic formula:
\[ x = \frac{-(-41) \pm \sqrt{1105}}{2 \times 24} = \frac{41 \pm \sqrt{1105}}{48} \]
Next, we can approximate the roots:
\[ \sqrt{1105} \approx 33.2 \quad \text{(approximately)} \] \[ x_1 \approx \frac{41 + 33.2}{48} \approx \frac{74.2}{48} \approx 1.55 \] \[ x_2 \approx \frac{41 - 33.2}{48} \approx \frac{7.8}{48} \approx 0.16 \]
Now, we can use these roots to analyze the function \(g(x)\). If you want to find the values of \(g(x)\) for specific \(x\) values in table 2, please specify the table entries or the functional form for \(g(x)\).
Let’s assume you want a corresponding \(g(x)\) value based on another quadratic or a linear function. If you have any specific relationship or function for \(g(x)\), please clarify so we can fill in table 2.