Which of the following quadratic equations can be solved by grouping?(1 point) Responses x2+10x+21=0 x squared plus 10 x plus 21 equals 0 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
The quadratic equation that can be solved by grouping is:
x2−12x+18=0
To determine which of the given quadratic equations can be solved by grouping, let's first understand what grouping means in this context.
Grouping is a method used to solve quadratic equations with four terms. The goal is to group the terms in a way that allows us to factor out a common binomial. By factoring out this common binomial, we can simplify the equation and solve for the variable.
Let's analyze the given quadratic equations one by one and see if they can be solved by grouping:
1. x^2 + 10x + 21 = 0:
This equation has three terms and does not fit the case for grouping because it lacks the required four terms.
2. x^2 - 4x - 8 = 0:
Similarly, this equation also has only three terms and does not meet the criteria for grouping.
3. x^2 + 8x - 22 = 0:
Once again, this equation consists of three terms, making it unsuitable for grouping.
4. x^2 - 12x + 18 = 0:
This equation has four terms and can be solved by grouping. Let's proceed with the process:
First, we split the middle term -12x into two terms such that their coefficients multiply to give the product of the coefficient of x^2 (which is 1) and the constant term 18. The product in this case is 1 * 18 = 18.
-12x = -6x - 6x [Splitting the -12x term: -6x * -6x = 36x^2]
Now we rewrite the equation with the grouped terms:
x^2 - 6x - 6x + 18 = 0
We can now factor by grouping:
(x^2 - 6x) - (6x - 18) = 0
x(x - 6) - 6(x - 3) = 0
(x - 6)(x - 3) = 0
By setting each factor equal to zero, we find the two solutions: x = 6 and x = 3.
In conclusion, out of the given quadratic equations, only x^2 - 12x + 18 = 0 can be solved by grouping.
To determine which of the given quadratic equations can be solved by grouping, let's analyze each equation:
1) x^2 + 10x + 21 = 0
This equation cannot be solved by grouping as the coefficients of x^2 and x are both positive.
2) x^2 - 4x - 8 = 0
This equation can be solved by grouping. We can rewrite it as:
(x^2 - 8x) + (4x - 8) = 0
x(x - 8) + 4(x - 2) = 0
Now, we can factor out the common terms:
(x + 4)(x - 2) = 0
So, the equation can be solved by grouping.
3) x^2 + 8x - 22 = 0
This equation cannot be solved by grouping as the middle term's coefficient is positive, while the constant term's coefficient is negative.
4) x^2 - 12x + 18 = 0
This equation cannot be solved by grouping as the middle term's coefficient is negative, while the constant term's coefficient is positive.
In conclusion, the quadratic equation that can be solved by grouping is x^2 - 4x - 8 = 0.