To determine which quadratic equation can be solved by grouping, we need to identify the quadratic that can be factored into a form where we can use grouping. Typically, in grouping, we look for a middle term that can be expressed as a sum of two numbers that multiply to the product of the leading coefficient (which is 1 in these cases) and the constant term.
Let's analyze each option:
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\(x^2 + 10x + 21 = 0\)
- Factors: \((x + 3)(x + 7) = 0\)
- Can be factored easily but does not use grouping.
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\(x^2 - 4x - 8 = 0\)
- The product of the leading coefficient (1) and the constant (-8) is -8. We need two numbers that multiply to -8 and add to -4. No such integers exist. Therefore, this one is not suitable for grouping.
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\(x^2 + 8x - 22 = 0\)
- The product of the leading coefficient (1) and the constant (-22) is -22. We need two numbers that multiply to -22 and add to 8. The factors that work here are (10, -2). Thus, this can be factored using grouping:
\[ x^2 + 10x - 2x - 22 = 0 \implies (x + 10)(x - 2) = 0 \] - This is a suitable candidate for grouping.
- The product of the leading coefficient (1) and the constant (-22) is -22. We need two numbers that multiply to -22 and add to 8. The factors that work here are (10, -2). Thus, this can be factored using grouping:
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\(x^2 - 12x + 18 = 0\)
- The product of the leading coefficient (1) and the constant (18) is 18; we need two numbers that multiply to 18 and add up to -12. The factors that work (if we were to find them) do not yield integers that fit being grouped well either.
Based on the analysis, the quadratic equation that can be solved by grouping is:
\(x^2 + 8x - 22 = 0\).
2 answers