To determine which of the given quadratic equations can be solved by grouping, we need to check if the middle term can be expressed as a sum of two terms whose product equals the product of the coefficient of the first term (which is 1) and the constant term.
Let's analyze each equation:
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\(x^2 + 10x + 21 = 0\)
- Factors: \(3\) and \(7\) (since \(3 + 7 = 10\) and \(3 \cdot 7 = 21\))
- This can be factored as \((x + 3)(x + 7) = 0\), so it can be solved by factoring, but not by grouping.
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\(x^2 + 8x - 22 = 0\)
- Factors: We need two numbers that add up to \(8\) and multiply to \(-22\). Those numbers are \(11\) and \(-2\).
- This can be factored as \((x + 11)(x - 2) = 0\), so it can also be solved by factoring, but not by grouping.
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\(x^2 - 12x + 18 = 0\)
- Factors: We need two numbers that add up to \(-12\) and multiply to \(18\). Those numbers are \(-6\) and \(-6\) (since \(-6 + -6 = -12\) and \(-6 \cdot -6 = 36\) which does not suit). You can't factor this using simple integers.
- Therefore, no easy factoring is available here.
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\(x^2 - 4x - 8 = 0\)
- Now, let's find the middle term. We need two numbers that add to \(-4\) and multiply to \(-8\). The numbers \( -6\) and \(2\) work (\(-6 + 2 = -4\) and \(-6 \cdot 2 = -12\)).
- However, since variables are involved in solving quadratics forms we cannot easily group.
From looking at the equations and performing checks, none of these equations fit the clean grouping method sequenced commonly in algebra, but if you have to choose based on their expressions fitted to simple factoring: We could still analyze them further with joining terms, but directly none 'fit' the typical grouping mechanism using numerical arrangement as specified for factory.
So, none of the responses here explicitly follow through the grouping method seamlessly but if you still wish grouping matters, come back to either of these: 1. 2.
Please note that the above analyses speak about solving via grouping hints, for educational purposes note interpretations change!
3 answers