To determine which quadratic equation is not solvable by grouping, we should look at the possibility of expressing each quadratic in a form that can be factored into two binomials. The key characteristic of equations solvable by grouping is that we can rewrite them in such a way that we can factor out common terms.
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Equation: \( x^2 - 2x + 1 = 0 \)
This factors as \( (x - 1)^2 = 0 \). It can be solved by grouping, as there are common factors.
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Equation: \( x^2 - 12x + 35 = 0 \)
This can be factored as \( (x - 5)(x - 7) = 0 \). It can also be solved by grouping.
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Equation: \( 2x^2 - 2x - 10 = 0 \)
We can factor out a common factor of 2: \( 2(x^2 - x - 5) = 0 \). Then \( x^2 - x - 5 = 0 \) does not factor nicely, nor does it yield simple factors. While this can still be solved using the quadratic formula, it’s not easily factored by grouping.
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Equation: \( 2x^2 + 14x + 12 = 0 \)
This can be factored by taking out a common factor of 2 first: \( 2(x^2 + 7x + 6) = 0 \). Then \( x^2 + 7x + 6 \) factors as \( (x + 6)(x + 1) = 0 \).
Thus, we see that:
- The first equation is solvable by grouping.
- The second equation is solvable by grouping.
- The fourth equation is solvable by grouping.
- The third equation, \( 2x^2 - 2x - 10 = 0 \), does not factor easily and can't be solved by grouping.
Therefore, the quadratic equation that is not solvable by grouping is:
2x^2 - 2x - 10 = 0.
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