Bot GPT 4-o mini To determine which of the given quadratic equations can be solved by grouping, we can look for pairs of numbers that multiply to give the constant term (the term without \(x\)) and add up to give the coefficient of the linear term (the \(x\) term).
Let's analyze each equation:
A. \(x^2 + 10x + 21 = 0\)
- Here, we need two numbers that multiply to \(21\) and add up to \(10\). The numbers \(3\) and \(7\) meet this condition:
- \(3 \times 7 = 21\)
- \(3 + 7 = 10\) Thus, we can factor this equation as \((x + 3)(x + 7) = 0\).
B. \(x^2 - 12x + 18 = 0\)
- We are looking for two numbers that multiply to \(18\) and add to \(-12\). The pairs of factors of \(18\) do not produce a pair that adds to \(-12\). Thus, this cannot be solved by grouping.
C. \(x^2 + 8x - 22 = 0\)
- Here, we need numbers that multiply to \(-22\) and add to \(8\). It is challenging to find such numbers, and thus this won't be suitable for grouping.
D. \(x^2 + 4x - 8 = 0\)
- We are looking for two numbers that multiply to \(-8\) and add to \(4\). However, similar to the above, no such pairs exist, indicating that grouping is not applicable here either.
In conclusion, the only equation that can be solved by grouping is:
A. \(x^2 + 10x + 21 = 0\).
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