viktorija
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3?
A. -6
B. -2
C. 2
D. 6
E. 12
Hello, everyone. I approached the question in a different manner from what I see posted above, so, in an effort to assist the community, I would like to share. We can rearrange the original equation to make everything easier to work with. Start by isolating the absolute value and then factor.
\(x^2 + 4x + 7 - (3) = |x + 2| + 3 - (3)\)
\(x^2 + 4x + 4 = |x + 2|\)
\((x + 2)^2 = |x + 2|\)
We do not need to go any further. We can appreciate that
the only time the value of an expression and its square will be the same is when the square equals 0 or 1. That leaves just three possibilities:
1) the expression equals 0
2) the expression equals -1
3) the expression equals 1
Thus, we need to work with x + 2 on its own.
\(x + 2 = 0\)
\(x = -2\) (solution 1)
\(x + 2 = -1\)
\(x = -3\) (solution 2)
\(x + 2 = 1\)
\(x = -1\) (solution 3)
Thus,
x can equal any of -1, -2, and -3. The product is a cinch:
\(-1 * -2 * -3 = -6\)
The answer must be (A). A little knowledge of number properties can go a long way here. Keep it simple. As always, good luck with your studies.
- Andrew