To determine which of the given quadratic equations is not solvable by grouping, we should consider their forms.
- \( x^2 - 2x + 1 = 0 \)
- \( 2x^2 - 2x - 10 = 0 \)
- \( 2x^2 + 14x + 12 = 0 \)
- \( x^2 - 12x + 35 = 0 \)
Grouping Method: This method involves rearranging and factoring the equation. Typically, it works well when the quadratic can be expressed in a way that allows us to group terms conveniently.
We'll start by analyzing each equation:
-
\( x^2 - 2x + 1 = 0 \)
- This factors easily as \( (x-1)^2 = 0 \).
-
\( 2x^2 - 2x - 10 = 0 \)
- Divide through by 2 to get \( x^2 - x - 5 = 0 \). This does not factor nicely, but it can be solved by completing the square or using the quadratic formula.
-
\( 2x^2 + 14x + 12 = 0 \)
- This can be factored as \( 2(x^2 + 7x + 6) = 0 \) and further factored as \( 2(x+6)(x+1) = 0 \), which can be grouped.
-
\( x^2 - 12x + 35 = 0 \)
- This factors to \( (x-5)(x-7) = 0 \), which can also be solved by grouping.
From this analysis, while all equations can be solved using various methods, the equation \( 2x^2 - 2x - 10 = 0 \) does not lend itself to “solving by grouping” because it requires simplification before any grouping strategies are applied.
Thus, the quadratic equation that is not solvable by grouping is: \( 2x^2 - 2x - 10 = 0 \).