Solve the following equation by grouping. x^2+10x+21=0 . Show all your work for full credit
To solve the equation x^2 + 10x + 21 = 0 by grouping, we need to find two numbers that add up to 10 (the coefficient of x) and multiply to 21 (the constant term).
The possible factors of 21 are 1, 3, 7, and 21. By trying different combinations, we can see that 3 and 7 are the numbers that satisfy both conditions.
Now, we rewrite the equation by breaking the middle term:
x^2 + 3x + 7x + 21 = 0
Next, we group the terms:
(x^2 + 3x) + (7x + 21) = 0
Now, we factor out the greatest common factor (GCF) from each group:
x(x + 3) + 7(x + 3) = 0
Notice that the terms (x + 3) are common in both groups. We can factor out this common binomial:
(x + 3)(x + 7) = 0
To solve the equation, we set each factor equal to zero and solve for x:
x + 3 = 0 or x + 7 = 0
For x + 3 = 0, we subtract 3 from both sides:
x = -3
For x + 7 = 0, we subtract 7 from both sides:
x = -7
Therefore, the solutions to the equation x^2 + 10x + 21 = 0 are x = -3 and x = -7.