Which of the following quadratic can be solved by grouping?
A. x^2 + 8x - 22 = 0
B. x^2 - 12x + 18 = 0
C. x^2 - 4x - 8 = 0
D. x^2 + 10x + 21 = 0
To solve a quadratic equation by grouping, one needs to factor the quadratic expression. Factoring the quadratic expression involves finding two binomials that multiply together to give the original quadratic expression.
Out of the given options, the quadratic expression that can be solved by grouping is the one that can be factored into two binomials.
A. x^2 + 8x - 22 = 0 - This quadratic expression cannot be factored into two binomials.
B. x^2 - 12x + 18 = 0 - This quadratic expression cannot be factored into two binomials.
C. x^2 - 4x - 8 = 0 - This quadratic expression cannot be factored into two binomials.
D. x^2 + 10x + 21 = 0 - This quadratic expression can be factored into (x + 3)(x + 7). Hence, it can be solved by grouping.
Therefore, the quadratic equation that can be solved by grouping is:
D. x^2 + 10x + 21 = 0
The quadratic equation that can be solved by grouping is:
C. x^2 - 4x - 8 = 0
To solve this quadratic equation by grouping, we need to find two numbers that multiply to give the constant term (-8) and add up to give the coefficient of the x term (-4).
The two numbers that satisfy this condition are -8 and 1, since (-8) * (1) = -8 and (-8) + (1) = -7.
Now, we can rewrite the quadratic equation as follows:
x^2 - 8x + x - 8 = 0
Next, we group the terms:
(x^2 - 8x) + (x - 8) = 0
Now, we factor out the common terms from each grouping:
x(x - 8) + 1(x - 8) = 0
We can see that there is a common factor of (x - 8) in both terms. Therefore, we can factor it out:
(x + 1)(x - 8) = 0
Finally, we set each factor equal to zero to find the solutions:
x + 1 = 0 or x - 8 = 0
Solving for x:
x = -1 or x = 8
Therefore, the quadratic equation x^2 - 4x - 8 = 0 can be solved by grouping, and the solutions are x = -1 and x = 8.
To determine which quadratic equation can be solved by grouping, we need to check if the equation can be factored using the grouping method. The grouping method involves splitting the middle term into two terms that can be factored separately.
Let's analyze each equation to see if it can be solved using the grouping method:
A. x^2 + 8x - 22 = 0
In this equation, the coefficient of x^2 is 1. The coefficient of x is positive (8), and the constant term is negative (-22). To factor this equation using grouping, we need to find two numbers whose product is -22 and sum is 8. However, there are no such numbers, so we cannot solve this equation by grouping.
B. x^2 - 12x + 18 = 0
In this equation, the coefficient of x^2 is 1. The coefficient of x is negative (-12), and the constant term is positive (18). To factor this equation using grouping, we need to find two numbers whose product is 18 and sum is -12. Again, there are no such numbers, so we cannot solve this equation by grouping.
C. x^2 - 4x - 8 = 0
In this equation, the coefficient of x^2 is 1. The coefficient of x is negative (-4), and the constant term is negative (-8). To factor this equation using grouping, we need to find two numbers whose product is -8 and sum is -4. The numbers that satisfy this are -8 and 1. Therefore, this equation can be factored by grouping:
(x - 8)(x + 1) = 0
So, equation C can be solved by grouping.
D. x^2 + 10x + 21 = 0
In this equation, the coefficient of x^2 is 1. The coefficient of x is positive (10), and the constant term is positive (21). To factor this equation using grouping, we need to find two numbers whose product is 21 and sum is 10. The numbers that satisfy this are 3 and 7. Therefore, this equation can be factored by grouping:
(x + 3)(x + 7) = 0
So, equation D can be solved by grouping.
In conclusion, the quadratic equations that can be solved by grouping are C (x^2 - 4x - 8 = 0) and D (x^2 + 10x + 21 = 0).