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

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

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

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

Answer 1

To determine which of the quadratic equations can be solved by grouping, we should look for equations where the middle term can be split into two terms whose products equal the product of the leading coefficient and the constant term.

Let’s analyze each equation:

1. $x^2-4x-8=0$

   • The product of the leading coefficient (1) and the constant term (-8) is -8. The factors of -8 that sum to -4 (the middle term) do not exist. Therefore, this equation cannot be solved by grouping.

2. $x^2-12x+18=0$

   • Here, the product is 1 * 18 = 18, and we are looking for two numbers that multiply to 18 and add up to -12. The factors -6 and -3 do not meet the requirement easily for grouping. Therefore, this may not be straightforward for grouping.

3. $x^2+8x-22=0$

   • The product is 1 * (-22) = -22. The factors that multiply to -22 and add to 8 do not exist (the possible factor pairs do not sum to 8). Therefore, this equation is also not suitable for grouping.

4. $x^2+10x+21=0$

   • The product is 1 * 21 = 21. The factors that multiply to 21 and add up to 10 are 3 and 7. Therefore, we can rewrite the equation as $x^2+3x+7x+21=0$ and factor by grouping: $(x^2+3x)+(7x+21)=0\to x(x+3)+7(x+3)=0\to (x+3)(x+7)=0$.

Based on this analysis, the quadratic equation that can be solved by grouping is:

$x^2+10x+21=0$.