Which of the following quadratic equations can be solved by grouping?
x^2−4x−8=0
x^2+10x+21=0
x^2−12x+18=0
x^2+8x−22=0
The quadratic equation that can be solved by grouping is:
x^2+10x+21=0
To solve a quadratic equation by grouping, we need to look for an equation in the form ax^2 + bx + c = 0 where the coefficient of x^2 is not 1. Let's check each of the given equations:
1. x^2 - 4x - 8 = 0: This equation has a coefficient of 1 for x^2, so we cannot solve it by grouping.
2. x^2 + 10x + 21 = 0: This equation has a coefficient of 1 for x^2, so we cannot solve it by grouping.
3. x^2 - 12x + 18 = 0: This equation does not have a coefficient of 1 for x^2, so we can solve it by grouping.
4. x^2 + 8x - 22 = 0: This equation has a coefficient of 1 for x^2, so we cannot solve it by grouping.
Therefore, the quadratic equation that can be solved by grouping is x^2 - 12x + 18 = 0.
To determine which of the given quadratic equations can be solved by grouping, we need to understand what it means to solve a quadratic equation by grouping.
When solving a quadratic equation by grouping, we typically have a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants. The goal is to group the terms in a way that allows us to factor the equation.
Let's apply this approach to each of the given quadratic equations:
1. x^2 - 4x - 8 = 0:
For this equation, grouping is not applicable. In this case, we can look for factors of -8 that combine to give -4 as the coefficient of the x term. However, no such factors exist.
2. x^2 + 10x + 21 = 0:
To determine if grouping works, we need to look for factors of 21 that combine to give 10 as the coefficient of the x term. Since 21 can be factored as 3 * 7, and 3 + 7 = 10, we can rewrite the equation as (x^2 + 3x) + (7x + 21) = 0. Now we can group the terms and factor them individually: x(x + 3) + 7(x + 3) = 0. Factoring out the common factor of (x + 3), we get (x + 3)(x + 7) = 0. Therefore, this equation can be solved by grouping.
3. x^2 - 12x + 18 = 0:
Similar to the previous equation, we need to determine if there are factors of 18 that combine to give -12 as the coefficient of the x term. However, there are no such factors. Hence, grouping is not applicable here.
4. x^2 + 8x - 22 = 0:
Again, we look for factors of -22 that combine to give 8 as the coefficient of the x term. However, there are no such factors. Therefore, grouping is not applicable to this equation either.
In conclusion, the only quadratic equation among the given options that can be solved by grouping is x^2 + 10x + 21 = 0.