viktorija wrote:
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
Some test-takers might find it easiest to proceed as follows:
1. Solve the equation with the signs unchanged
2. Solve the equation with the signs changed in the absolute value
In each case, plug any possible solutions back into the original equation to confirm that they are valid.
\(x^2 + 4x + 7 = |x + 2| + 3\)
\(x^2 + 4x + 4 = |x + 2|\)
Here, a solution will be valid as along as the left side is NONNEGATIVE.
Case 1: signs unchanged\(x^2 + 4x + 4 = x + 2\)
\(x^2 + 3x + 2 = 0\)
\((x+2)(x+1) = 0\)
Possible solutions: x=-2, x=-1.
If we plug \(x=-2\) into \(x^2 + 4x + 4\), we get:
\((-2)^2 + (4)(-2) + 4 = 4 - 8 + 4 = 0\).
Since the result is nonnegative, x=-2 is a valid solution.
If we plug \(x=-1\) into \(x^2 + 4x + 4\), we get:
\((-1)^2 + (4)(-1) + 4 = 1 - 4 + 4 = 1\).
Since the result is nonnegative, x=-1 is a valid solution.
Case 2: signs changed in the absolute value\(x^2 + 4x + 4 = -x - 2\)
\(x^2 + 5x + 6 = 0\)
\((x+3)(x+2) = 0\)
Possible solutions: x=-3, x=-2.
Since x=-2 was a valid solution in Case 1, only x=-3 must be tested.
If we plug \(x=-3\) into \(x^2 + 4x + 4\), we get:
\((-3)^2 + (4)(-3) + 4 = 9 - 12 + 4 = 1\).
Since the result is nonnegative, x=-3 is a valid solution.
Product of the solutions = (-2)(-1)(-3) = -6.
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