To determine which of the given quadratic equations is not solvable by grouping, we first need to understand what grouping entails. In general, grouping is a method used to factor a polynomial by grouping the terms into pairs or groups that can be factored out.
Let's analyze each of the quadratic equations provided:
- \(2x^2 + 14x + 12 = 0\)
This can be factored by grouping. First, factor out the GCF (2): \[ 2(x^2 + 7x + 6) = 0 \] Then, we can factor the quadratic inside the parentheses: \[ x^2 + 7x + 6 = (x + 6)(x + 1) \]
- \(x^2 - 12x + 35 = 0\)
This equation can also be factored. It factors directly: \[ (x - 5)(x - 7) = 0 \] Grouping is not necessary here, but it can still be considered solvable.
- \(2x^2 - 2x - 10 = 0\)
First, divide the entire equation by 2: \[ x^2 - x - 5 = 0 \] This does not factor neatly, and thus cannot be solved by grouping.
- \(x^2 - 2x + 1 = 0\)
This can be factored straightforwardly: \[ (x - 1)(x - 1) = 0 \]
From the analysis, the equation \(2x^2 - 2x - 10 = 0\) is the one that does not factor neatly. Therefore, it cannot be solved by grouping as effectively as the others.
Answer: \(2x^2 - 2x - 10 = 0\) is not solvable by grouping.
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