To determine which of the given quadratic equations can be solved by grouping, we need to check if the middle term can be split into two terms that allow for a common factor extraction.
-
\(x^2 - 12x + 18 = 0\): The constant term 18 doesn't factor nicely to split -12 into two integers. Not suitable for grouping.
-
\(x^2 + 8x - 22 = 0\): The constant term -22 also doesn't allow for a split of 8 into two factors. Not suitable for grouping.
-
\(x^2 - 4x - 8 = 0\): The constant term -8 could potentially split -4, but it’s not straightforward to find two factors that work. Also, not suitable for grouping.
-
\(x^2 + 10x + 21 = 0\): The middle term 10 can be split into 3 and 7 (since 3 * 7 = 21, and 3 + 7 = 10).
Thus, the equation \(x^2 + 10x + 21 = 0\) can be solved by grouping, as it can be written as:
\[ x^2 + 3x + 7x + 21 = 0 \implies x(x + 3) + 7(x + 3) = 0 \implies (x + 3)(x + 7) = 0 \]
So the answer is:
\(x^2 + 10x + 21 = 0\)