Which of the following quadratic equations can be solved by grouping?
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+10x+21=0 x squared plus 10 x plus 21 equals 0 x2+8x−22=0
The quadratic equation that can be solved by grouping is:
x² - 12x + 18 = 0
The quadratic equation that can be solved by grouping is:
x² - 12x + 18 = 0 (x squared minus 12 x plus 18 equals 0)
To solve this equation by grouping, follow these steps:
Step 1: Split the middle term (-12x) into two terms such that their sum is equal to -12x and their product is equal to the product of the first and last terms. In this case, the first term is x² and the last term is 18.
-12x can be split as -6x - 6x, because -6x + (-6x) = -12x and (-6x) * (-6x) = 36x².
Step 2: Group the terms.
x² - 6x - 6x + 18 = 0
Step 3: Factor by grouping.
(x² - 6x) + (-6x + 18) = 0
Step 4: Factor each group separately.
x(x - 6) - 6(x - 6) = 0
Step 5: Factor out the common factor (x - 6).
(x - 6)(x - 6) = 0
Step 6: Set each factor equal to zero and solve for x.
x - 6 = 0
x = 6
Therefore, the solution to the quadratic equation x² - 12x + 18 = 0 is x = 6.
To determine which of the given quadratic equations can be solved by grouping, we need to understand the concept of grouping in solving quadratic equations.
Grouping is a technique used to solve a quadratic equation when the middle term can be split into two terms such that it can be factored by grouping. In other words, it involves finding two numbers that multiply to give the product of the coefficient of the squared term and the constant term, and also add up to give the coefficient of the middle term.
Let's analyze each equation and determine if it can be solved by grouping:
1) x^2 - 4x - 8 = 0:
This equation cannot be solved by grouping as the middle term (-4x) does not have two numbers that can be added to give -4 and multiplied to give -8.
2) x^2 - 12x + 18 = 0:
This equation can be solved by grouping. The middle term (-12x) can be split into -6x and -6x. Then, we can factor by grouping:
x^2 - 12x + 18 = (x^2 - 6x) + (-6x + 18) = x(x - 6) - 6(x - 6) = (x - 6)(x - 6).
3) x^2 + 10x + 21 = 0:
This equation cannot be solved by grouping because the middle term (10x) does not have two numbers that can be added to give 10 and multiplied to give 21.
4) x^2 + 8x - 22 = 0:
This equation can be solved by grouping. The middle term (8x) can be split into 11x and -3x. Then, we can factor by grouping:
x^2 + 8x - 22 = (x^2 + 11x) + (-3x - 22) = x(x + 11) - 3(x + 11) = (x - 3)(x + 11).
Based on the analysis, the quadratic equations that can be solved by grouping are:
1) x^2 - 12x + 18 = 0
2) x^2 + 8x - 22 = 0