To solve the equation \(x^2 - x - 6 = 0\) by inspection, we can factor it. We need to find two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \(x\)).
The numbers that meet these criteria are \(2\) and \(-3\) because:
- \(2 \times -3 = -6\)
- \(2 + (-3) = -1\)
This means we can factor the equation as: \[(x - 3)(x + 2) = 0\]
Setting each factor to zero gives us the solutions:
- \(x - 3 = 0 \implies x = 3\)
- \(x + 2 = 0 \implies x = -2\)
However, the provided options were:
- \(x = -6\)
- \(x = -1\)
- \(x = 1\)
- \(x = 6\)
It seems that the numbers \(x=3\) and \(x=-2\) aren't among the options, indicating that we might have been misreading the question or options.
None of these numbers are solutions to the equation \(x^2 - x - 6 = 0\). However, if we strictly consider the task to identify options close to the correct numbers or check intuitively, we see that neither solution, \(3\) or \(-2\), finds footing in the presented options.
To accurately respond to which selections might draw any overlap or match mimic what fits best within the provided scheme, we can ponder the intersections of roots and weave through misalignment, but robust selection is elusive against provided choices.
To conclude, based on the potential glare: None of the responses provided are correct solutions to the equation \(x^2 - x - 6 = 0\).
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