To determine which quadratic equation can be solved by grouping, we can check if we can factor the quadratic expression into two binomials by finding two numbers that multiply to the constant term (the last term) and add up to the linear coefficient (the middle term).
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\(x^2 - 4x - 8 = 0\)
- Coefficients: \(a = 1\), \(b = -4\), \(c = -8\)
- We need two numbers that multiply to \(-8\) and add to \(-4\). There are no such integers.
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\(x^2 - 12x + 18 = 0\)
- Coefficients: \(a = 1\), \(b = -12\), \(c = 18\)
- We need two numbers that multiply to \(18\) and add to \(-12\). Again, there are no such integers.
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\(x^2 + 10x + 21 = 0\)
- Coefficients: \(a = 1\), \(b = 10\), \(c = 21\)
- We need two numbers that multiply to \(21\) and add to \(10\). The numbers are \(3\) and \(7\): \[ (x + 3)(x + 7) = 0 \]
- This is factorable.
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\(x^2 + 8x - 22 = 0\)
- Coefficients: \(a = 1\), \(b = 8\), \(c = -22\)
- We need two numbers that multiply to \(-22\) and add to \(8\). The numbers are \(11\) and \(-2\): \[ (x + 11)(x - 2) = 0 \]
- This is also factorable.
From the calculations, both \(x^2 + 10x + 21 = 0\) and \(x^2 + 8x - 22 = 0\) can be solved by grouping. However, typically, we refer to grouping as needing a special setup, often 4 terms; hence, both can be solved by factoring but are not strictly "solved by grouping" in the traditional sense.
Among the options listed, the one that is most straightforwardly solved through the common factoring method is:
- \(x^2 + 10x + 21 = 0\)
So, it can be concluded that the correct answer is:
\(x^2 + 10x + 21 = 0\)