Question

Which of the following quadratic equations can be solved by grouping?
A. X^2 - 12x + 18 = 0
B. X^2 - 4x - 8 = 0
C. X^2 + 8x - 22=0
D. X^2 + 10x + 21 = 0

Answer 1

To determine which quadratic equations can be solved by grouping, we look for a way to factor them into a product of binomials that can be grouped effectively. Typically, we would look for a quadratic equation that can be rearranged into a form that allows us to factor it by grouping the terms.

Let’s analyze each quadratic equation one by one:

**A.** \(X^2 - 12x + 18 = 0\)
The factors of 18 that add up to -12 do not exist. It does not appear to be easily factored by grouping.

**B.** \(X^2 - 4x - 8 = 0\)
We need to find two numbers that multiply to -8 and add to -4. The numbers -6 and +2 work, hence we can factor this as:
\((X - 6)(X + 2) = 0\)
This can be solved by grouping since we can group terms after rearranging.

**C.** \(X^2 + 8x - 22 = 0\)
The factors of -22 that would add up to +8 do not exist. It is not suitable for grouping.

**D.** \(X^2 + 10x + 21 = 0\)
This can be factored directly as:
\((X + 3)(X + 7) = 0\)
But it does not require grouping as it can be factored directly.

After careful consideration, **Option B** (\(X^2 - 4x - 8 = 0\)) is the only quadratic equation that can be solved effectively by grouping.