Which of the following quadratic equations can be solved by grouping?(1 point)

Question

Responses

x2−4x−8=0
x squared minus 4 x minus 8 equals 0

x2+8x−22=0
x squared plus 8 x minus 22 equals 0

x2−12x+18=0
x squared minus 12 x plus 18 equals 0

x2+10x+21=0
x squared plus 10 x plus 21 equals 0

Answer 1

To determine which of the given quadratic equations can be solved by grouping, we need to factor each equation.

Grouping is useful for quadratic equations when they can be expressed as a product of two binomials. This usually requires the middle coefficient to be expressible as a sum of two numbers which multiply to the product of the first and last coefficients.

Let's analyze the equations one by one:

1. \(x^2 - 4x - 8 = 0\):

   • The factors of -8 that sum to -4 do not exist. This cannot be solved by grouping.

2. \(x^2 + 8x - 22 = 0\):

   • The factors of -22 that sum to 8 do not exist. This cannot be solved by grouping.

3. \(x^2 - 12x + 18 = 0\):

   • The factors of 18 that sum to -12 are -6 and -6. This can be factored as \((x - 6)(x - 6) = 0\) and can also be approached by grouping.

4. \(x^2 + 10x + 21 = 0\):

   • The factors of 21 that sum to 10 are 3 and 7. This can be factored as \((x + 3)(x + 7) = 0\) and can be approached by grouping.

Based on this analysis, the equations that can be solved by grouping are:

• \(x^2 - 12x + 18 = 0\)
• \(x^2 + 10x + 21 = 0\)

However, the question asks for one of the options that can be solved by grouping, and both of these options can serve that purpose. If you need to select one, \(x^2 - 12x + 18 = 0\) might be a straightforward answer due to it having repeated roots, but you'll want to select based on the instruction or your specific context.