To determine which of the given quadratic equations can be solved by grouping, we need to look for a pair of terms that can be factored in such a way that we can group them effectively.
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a: \(x^2 + 8x - 22 = 0\)
This does not factor nicely and does not lend itself to grouping. -
b: \(x^2 + 10x + 21 = 0\)
This can be factored as \((x + 3)(x + 7) = 0\). However, it does not require grouping. -
c: \(x^2 - 12x + 18 = 0\)
This can be factored as \((x - 6)(x - 3) = 0\). Again, it can be factored easily but does not show a typical grouping scenario. -
d: \(x^2 - 4x - 8 = 0\)
We can rewrite this as \(x^2 - 4x + 4 - 12 = 0\), leading to \( (x - 2)^2 - 12 = 0\). However, it can also be rearranged and factored, but it is less straightforward for grouping.
Among the options, none explicitly lend themselves clearly and directly to the method of grouping that we typically see used for cubics or higher degree polynomials.
However, for the purpose of identifying potential grouping, you may want to observe how you might create a pair:
- For d, reordering to bring it into a form like \(x^2 - 4x + 4 - 12 = 0\) does allow for some grouping if we consider the constant transformation.
Based on these considerations, if any of these were to align most closely with grouping, it would be option d, although this isn't a classic example of grouping as seen with higher-degree polynomials. The primary goal would still be to check possible grouping frames in these factorizations.
So, if you're strictly looking for classic grouping, the option that comes closest to exhibiting that form in modification would be d, \(x^2 - 4x - 8 = 0\).