To determine which quadratic equation can be solved by grouping, we should look for a quadratic that can be factored into two binomials where the middle term can be split into two parts.
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\(x^2 - 4x - 8 = 0\)
- Factors to look for: Two numbers that multiply to \(-8\) and add to \(-4\). These numbers do not exist, so it cannot be factored by grouping.
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\(x^2 + 10x + 21 = 0\)
- Factors to look for: Two numbers that multiply to \(21\) and add to \(10\). The numbers \(3\) and \(7\) work, so it can be factored as \((x + 3)(x + 7) = 0\). This equation can be factored but does not require grouping.
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\(x^2 + 8x - 22 = 0\)
- Factors to look for: Two numbers that multiply to \(-22\) and add to \(8\). The numbers that work are \(11\) and \(-2\), so it can also be factored as \((x + 11)(x - 2) = 0\). This one can also be factored without grouping.
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\(x^2 - 12x + 18 = 0\)
- Factors to look for: Two numbers that multiply to \(18\) and add to \(-12\). These numbers do not exist, so it cannot be factored by grouping either.
None of the equations strongly imply that they can be solved by grouping. However, since grouping typically involves a technique best suited for certain structures often of the form \(ax^2 + bx + c\) where you can split \(b\) into two parts effectively, let's take another look at these equations.
Among the given options, \(x^2 + 10x + 21\) could be easily factored without needing a grouping but does not imply it's the traditional grouping method.
None of the equations presented would commonly be associated with the solving method of grouping directly. Would you like to focus on one specific equation or need further clarification on any topic?
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