Bot GPT 4-o mini Let the whole number be denoted by \( x \). According to the problem, we have the equation based on the statement:
\[ 3x^2 - 2x = 133 \]
To rearrange this, we can move all terms to one side:
\[ 3x^2 - 2x - 133 = 0 \]
Now we can solve this quadratic equation using the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 3 \), \( b = -2 \), and \( c = -133 \). Substituting these values into the formula gives:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-133)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ (-2)^2 - 4 \cdot 3 \cdot (-133) = 4 + 1596 = 1600 \]
Now substituting this back into the quadratic formula:
\[ x = \frac{2 \pm \sqrt{1600}}{6} \]
Calculating \( \sqrt{1600} \):
\[ \sqrt{1600} = 40 \]
So we have:
\[ x = \frac{2 \pm 40}{6} \]
This gives us two possible solutions:
- \( x = \frac{42}{6} = 7 \)
- \( x = \frac{-38}{6} = -\frac{19}{3} \) (which is not a whole number)
Thus, the only whole number solution is:
\[ \boxed{7} \]
To verify, we can substitute \( x = 7 \) back into the original statement:
Calculating \( 3x^2 \):
\[ 3(7^2) = 3 \times 49 = 147 \]
Calculating \( 2x \):
\[ 2 \times 7 = 14 \]
Subtracting these gives:
\[ 147 - 14 = 133 \]
Since this holds true, the solution \( x = 7 \) is correct.
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