Which of the following quadratic equations can be solved by grouping?(1 point)
Responses
x2+8x−22=0
x squared plus 8 x minus 22 equals 0
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
The method of solving quadratic equations by grouping is often used when you can factor the quadratic into a product of two binomials. Let's see which of the given options can be factored this way:
1. \( x^2 + 8x - 22 = 0 \)
For this equation, we need to find two numbers that multiply to -22 and add up to 8. There are no such integers that fulfill both conditions, so this cannot be factored easily by grouping.
2. \( x^2 - 4x - 8 = 0 \)
Similarly, we are looking for two numbers that multiply to -8 and add up to -4. Again, there are no such integers that fulfill these conditions, so this cannot be factored easily by grouping either.
3. \( x^2 - 12x + 18 = 0 \)
For this quadratic equation, we are looking for two numbers that multiply to 18 and add up to -12. The numbers that satisfy these criteria are -6 and -6, since (-6) * (-6) = 18 and (-6) + (-6) = -12. Therefore, this equation can be factored into:
\( (x - 6)(x - 6) = 0 \)
This equation can be solved by grouping.
4. \( x^2 + 10x + 21 = 0 \)
In this equation, we are looking for two numbers that multiply to 21 and add up to 10. The numbers that fulfill both conditions are 3 and 7, since 3 * 7 = 21 and 3 + 7 = 10. So, this equation can be factored into:
\( (x + 3)(x + 7) = 0 \)
This equation can also be solved by grouping.
In summary, the quadratic equations that can be solved by grouping are:
\( x^2 - 12x + 18 = 0 \)
\( x^2 + 10x + 21 = 0 \)