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Bot GPT 4-o mini To determine which quadratic equation can be solved by grouping, we typically look for a way to factor the quadratic into two binomials. Solving by grouping often requires the middle term to be split into two parts whose products give the constant term when multiplied.
Let’s examine each equation:
1. **\( x^2 - 12x + 18 = 0 \)**
- Factors of \(18\) that add up to \( -12\): \( -6 \) and \( -6 \).
- This can be factored as \( (x - 6)(x - 6) = 0 \). Not typically solved by grouping.
2. **\( x^2 - 4x - 8 = 0 \)**
- Factors of \(-8\) that add up to \(-4\) do not exist. This cannot be factored nicely; grouping is not applicable.
3. **\( x^2 + 8x - 22 = 0 \)**
- Factors of \(-22\) that add up to \(8\):
- It does factor as \( (x + 11)(x - 2) = 0 \), but doesn’t show a clear grouping approach.
4. **\( x^2 + 10x + 21 = 0 \)**
- Factors of \(21\) that add up to \(10\): \(3\) and \(7\).
- This can be factored as \( (x + 3)(x + 7) = 0\), which is straightforward and not typically solved by grouping.
None of these quadratic equations appear to be particularly suited for grouping. However, the second equation (\( x^2 - 4x - 8 = 0 \)) might be attempted through grouping or rearranging, despite not being straightforward.
Therefore, if choosing strictly based on the potential for grouping, option (2) \( x^2 - 4x - 8 = 0 \) might be what you are looking for, but others also could be manipulated depending on the method applied.
In summary: **None accurately fall into a standard 'grouping' technique for simple solving.** Nevertheless, the closest would be \( x^2 - 4x - 8 = 0 \).