To determine which quadratic equation can be solved by grouping, we generally look for a quadratic expression that can be factored by grouping techniques. This often occurs when there are factors of the quadratic that can be combined to form two groups that share a common factor.
Let's analyze the options provided:
-
\( x^2 + 10x + 21 = 0 \): This factors as \( (x + 3)(x + 7) = 0 \). It does not require grouping.
-
\( x^2 - 4x - 8 = 0 \): This can be factored as \( (x - 6)(x + 2) = 0 \) or can be solved using the quadratic formula but does not directly lend itself to grouping.
-
\( x^2 + 8x - 22 = 0 \): This does not easily factor. It can also be solved using the quadratic formula but isn't suitable for grouping.
-
\( x^2 - 12x + 18 = 0 \): This can be transformed to \( x^2 - 12x + 36 - 18 = 0\), recognizing that \(x^2 - 12x + 36 = (x - 6)^2\), thus might be bridged to traditional grouping options. However, it's more common to use square completion.
Among these equations, none lend themselves perfectly to the typical "grouping" method. However, quadratic equations that can be factored into sets where pairings produce common terms are more optimal candidates for grouping; typically, you would want the middle term to be expressible as two factors that can lead to grouping.
Given these considerations, in the absence of a perfect candidate, one might lean towards the equation that requires factoring expression through recognizable pair grouping (though it's not explicit here), hence the relative candidate if forced into choice could be \( x^2 - 12x + 18 = 0 \) due to the potential reformation into recognizable standard.
Ultimately, none of the equations perfectly fit typical grouping but if choosing based on factor factorization styles, then \(x^2 - 12x + 18 = 0\) resonates best.
2 answers