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
A. X^2 - 12x + 18 = 0
B. X^2 - 4x - 8 = 0
C. X^2 + 8x - 22 = 0
Answers
GPT-4o mini
To determine which of the given quadratic equations can be solved by grouping, we generally look for a way to express the quadratic in a form that allows us to factor it into two binomials.
Let's consider each equation:
A. \( x^2 - 12x + 18 = 0 \)
The quadratic can be rewritten by finding two numbers that multiply to \( +18 \) (the constant term) and add to \( -12 \) (the coefficient of the linear term). The numbers \( -6 \) and \( -6 \) multiply to \( 36 \) and add to \( -12 \), but they do not lead to the correct factorization since \( 18 \) doesn’t have such pairs. This equation is not easily factored by grouping.
B. \( x^2 - 4x - 8 = 0 \)
Similarly, we need two numbers that multiply to \( -8 \) and add to \( -4 \). The numbers \( 2 \) and \( -4 \) multiply to \( -8 \) and add to \( -2 \), while \( -6 \) and \( 2 \) could add as needed but don't multiply correctly. Again, this equation does not give a good grouping.
C. \( x^2 + 8x - 22 = 0 \)
We find two numbers that multiply to \( -22 \) and add to \( +8 \). The numbers \( 11 \) and \( -2 \) work since \( 11 \times (-2) = -22 \) and \( 11 + (-2) = 9 \), but again, we can’t group these to factor nicely.
Actually, we can realize that in B \(x^2 - 4x - 8\) can be factored and rearranged for dividing and solving as a pattern of grouping.
Out of the three options, B ( \( x^2 - 4x - 8 = 0 \)) is the most amenable to solving by grouping, as it can be rearranged in different ways to reveal manageable components to calculate roots thereof.
Thus, the answer is:
**B. \( x^2 - 4x - 8 = 0 \)**
Let's consider each equation:
A. \( x^2 - 12x + 18 = 0 \)
The quadratic can be rewritten by finding two numbers that multiply to \( +18 \) (the constant term) and add to \( -12 \) (the coefficient of the linear term). The numbers \( -6 \) and \( -6 \) multiply to \( 36 \) and add to \( -12 \), but they do not lead to the correct factorization since \( 18 \) doesn’t have such pairs. This equation is not easily factored by grouping.
B. \( x^2 - 4x - 8 = 0 \)
Similarly, we need two numbers that multiply to \( -8 \) and add to \( -4 \). The numbers \( 2 \) and \( -4 \) multiply to \( -8 \) and add to \( -2 \), while \( -6 \) and \( 2 \) could add as needed but don't multiply correctly. Again, this equation does not give a good grouping.
C. \( x^2 + 8x - 22 = 0 \)
We find two numbers that multiply to \( -22 \) and add to \( +8 \). The numbers \( 11 \) and \( -2 \) work since \( 11 \times (-2) = -22 \) and \( 11 + (-2) = 9 \), but again, we can’t group these to factor nicely.
Actually, we can realize that in B \(x^2 - 4x - 8\) can be factored and rearranged for dividing and solving as a pattern of grouping.
Out of the three options, B ( \( x^2 - 4x - 8 = 0 \)) is the most amenable to solving by grouping, as it can be rearranged in different ways to reveal manageable components to calculate roots thereof.
Thus, the answer is:
**B. \( x^2 - 4x - 8 = 0 \)**