To determine which of the given quadratic equations can be solved by grouping, we need to factor the quadratic expressions. Grouping usually works when we can rewrite the quadratic into a form that allows us to factor it into two binomials.
Let's analyze each equation:
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\(x^2 + 10x + 21 = 0\):
- Factors to \((x + 3)(x + 7) = 0\)
- This can be solved by factoring directly, not by grouping.
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\(x^2 - 4x - 8 = 0\):
- The factors do not come out to be nice integers. It could be solved using the quadratic formula, but it does not lend itself to grouping.
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\(x^2 + 8x - 22 = 0\):
- This doesn't factor nicely. The sum of factors needs to add up to 8 and multiply to -22, which does not yield integer factors. Moreover, it does not fit into a grouping framework.
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\(x^2 - 12x + 18 = 0\):
- To check if this can be grouped, we can look for two numbers that multiply to \(18\) and add to \(-12\).
- This can't be easily factored with integers into a form that allows grouping.
None of these equations are suitable for grouping as none produce a factorizable expression conducive to that method. However, if we do look for one that might work with clever rearranging, the answer is none of them go directly into a useful method of grouping based on initial inspection. If you must select one, it would be ideal to check each through numerical estimates or attempts to reorganize terms creatively, but in standard application, none present immediate groupable traits.
Nonetheless, it seems like usually we'd consider facilitating factorization methods like completing the square or quadratic formula for these equations, confirming neither applied standardly fits grouping criteria.
That said, the answer for selection, given strict criteria, tends to be that most normal quadratic scenarios publicly yield the dame approaches (i.e. factoring direct/hands-off). All options generally present nil solutions there. I guess if pressed strictly, you might argue they all fail such grouping against quadratic standards here.
3 answers