To determine which of the given quadratic equations is not solvable by grouping, we can first look at the possibility of factoring each equation. Grouping typically involves factoring a quadratic into two binomials.
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\(2x^2 - 2x - 10 = 0\):
- This can be factored as \(2(x^2 - x - 5) = 0\) and further factoring is not easy, so it might not be solved by simple grouping.
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\(x^2 - 2x + 1 = 0\):
- This can be factored as \((x-1)^2 = 0\), which is easily solvable.
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\(x^2 - 12x + 35 = 0\):
- This can be factored as \((x-5)(x-7) = 0\), again easily solvable.
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\(2x^2 + 14x + 12 = 0\):
- This can be factored by first taking out a common factor of 2: \(2(x^2 + 7x + 6) = 0\), which can be factored further into \(2(x+6)(x+1) = 0\).
From the analysis above, the equation \(2x^2 - 2x - 10 = 0\) is harder to factor using grouping compared to the other quadratic equations, which can be factored easily. Therefore, the correct answer is:
\(2x^2 - 2x - 10 = 0\) is the quadratic equation that is not solvable by grouping.