Solve the following equation by grouping. x2+10x+21=0 . Show all your work for full credit. (2 points)
To solve the equation x^2 + 10x + 21 = 0 by grouping, we need to find two numbers whose sum is 10 and whose product is 21.
Let's factor the constant term, 21:
21 = 1 × 21
= 3 × 7
Now, we need to find two numbers whose sum is 10. The only combination of these factors that will work is 3 and 7:
3 + 7 = 10
So, we can rewrite the middle term, 10x, as 3x + 7x:
x^2 + 3x + 7x + 21 = 0
Now, let's group the terms:
(x^2 + 3x) + (7x + 21) = 0
Next, we can factor out the greatest common factor (GCF) from each group:
x(x + 3) + 7(x + 3) = 0
Now, we see that we have a common binomial factor, (x + 3), in both terms. We can factor it out:
(x + 3)(x + 7) = 0
The equation is now factored. To solve for x, we set each binomial factor equal to zero:
x + 3 = 0 or x + 7 = 0
For the first equation, subtracting 3 from both sides gives:
x = -3
For the second equation, subtracting 7 from both sides gives:
x = -7
Therefore, the solutions to the equation x^2 + 10x + 21 = 0 are x = -3 and x = -7.