To determine which of the given quadratic equations is not solvable by grouping, we can look for expressions where the middle term cannot be rewritten as the sum of two terms whose products yield the product of the leading coefficient and the constant term.
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\(x^2 - 2x + 1 = 0\)
- This can be factored as \((x - 1)(x - 1) = 0\).
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\(2x^2 - 2x - 10 = 0\)
- We can simplify by dividing the entire equation by 2: \(x^2 - x - 5 = 0\). This does not factor neatly, but it can still be solved using the quadratic formula.
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\(x^2 - 12x + 35 = 0\)
- This can be factored as \((x - 5)(x - 7) = 0\).
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\(2x^2 + 14x + 12 = 0\)
- We can factor this by first dividing everything by 2: \(x^2 + 7x + 6 = 0\). This can be factored as \((x + 1)(x + 6) = 0\).
Among these, the equation \(2x^2 - 2x - 10 = 0\) is the most complicated in terms of straightforward factoring. It can be solved through the quadratic formula but doesn't factor easily (no integers exist that fit into the grouping method).
Therefore, the quadratic equation that is not easily solvable by grouping is:
2x^2 − 2x − 10 = 0.