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What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3
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21 Nov 2021, 05:16
21 Nov 2021, 05:16
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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
A. -6
B. -2
C. 2
D. 6
E. 12
Re: What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3
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22 Nov 2021, 02:59
22 Nov 2021, 02:59
2
\(x^2 + 4x + 7 = \textbf{|x+2|} + 3\)
When \(|x+2| = +(x+2)\),
\(x^2 + 4x + 7 = x + 2 + 3\)
\(x^2 + 3x + 2 = 0\)
\(x^2 + (2+1)x + 2*1 = 0 \Rightarrow x = -1\) and \(-2\)
When \(|x+2| = -(x+2)\),
\(x^2 + 4x + 7 = -x - 2 + 3\)
\(x^2 + 5x + 6 = 0\)
\(x^2 + (3+2)x + 3*2 = 0 \Rightarrow x = -3\) and \(-2\)
Therefore, possible values of \(x = -1, -2\) and \(-3\)
Therefore, product of all solutions to the equations = \((-1)*(-2)*(-3) = \textbf{-6}\)
Hence, Answer is A
When \(|x+2| = +(x+2)\),
\(x^2 + 4x + 7 = x + 2 + 3\)
\(x^2 + 3x + 2 = 0\)
\(x^2 + (2+1)x + 2*1 = 0 \Rightarrow x = -1\) and \(-2\)
When \(|x+2| = -(x+2)\),
\(x^2 + 4x + 7 = -x - 2 + 3\)
\(x^2 + 5x + 6 = 0\)
\(x^2 + (3+2)x + 3*2 = 0 \Rightarrow x = -3\) and \(-2\)
Therefore, possible values of \(x = -1, -2\) and \(-3\)
Therefore, product of all solutions to the equations = \((-1)*(-2)*(-3) = \textbf{-6}\)
Hence, Answer is A
General Discussion
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What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3
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30 Dec 2021, 21:31
30 Dec 2021, 21:31
Given that \(x^2 + 4x + 7 = |x + 2| + 3\) and we need to find the product of all the solutions of this equation
To open the Absolute Value we need to consider two cases (Watch this video to learn about the Basics of Absolute Value)
Case 1: x+2 >= 0 or x >= -2
=> |x+2| = x+2 (as x+2 is non-negative)
=> \(x^2 + 4x + 7 = |x + 2| + 3\) = \(x^2 + 4x + 7 = x+2 + 3\)
=> \(x^2 + 4x - x + 7 - 5\) = 0
=> \(x^2 + 3x + 2 \) = 0
=> \(x^2 + x + 2x + 2 \) = 0
=> x(x+1) + 2(x+1) = 0
=> (x+1) (x+2) = 0
=> x = -1 , -2
Both the solutions are in the range x >= -2, so are true
Case 2: x+2 < 0 or x < -2
=> |x+2| = -(x+2) (as x+2 is negative)
=> \(x^2 + 4x + 7 = |x + 2| + 3\) = \(x^2 + 4x + 7 = -(x+2) + 3\)
=> \(x^2 + 4x + x + 7 - 1\) = 0
=> \(x^2 + 5x + 6 \) = 0
=> \(x^2 + 2x + 3x + 6 \) = 0
=> x(x+2) + 3(x+2) = 0
=> (x+2) (x+3) = 0
=> x = -2 , -3
But the range was x < -2, so x = -3 is the solution in this case
=> x = -3, -2, -1
=> Product of the solutions = -3*-2*-1 = -6
So, Answer will be A
Hope it helps!
Watch the following video to learn How to Solve Absolute Value Problems
To open the Absolute Value we need to consider two cases (Watch this video to learn about the Basics of Absolute Value)
Case 1: x+2 >= 0 or x >= -2
=> |x+2| = x+2 (as x+2 is non-negative)
=> \(x^2 + 4x + 7 = |x + 2| + 3\) = \(x^2 + 4x + 7 = x+2 + 3\)
=> \(x^2 + 4x - x + 7 - 5\) = 0
=> \(x^2 + 3x + 2 \) = 0
=> \(x^2 + x + 2x + 2 \) = 0
=> x(x+1) + 2(x+1) = 0
=> (x+1) (x+2) = 0
=> x = -1 , -2
Both the solutions are in the range x >= -2, so are true
Case 2: x+2 < 0 or x < -2
=> |x+2| = -(x+2) (as x+2 is negative)
=> \(x^2 + 4x + 7 = |x + 2| + 3\) = \(x^2 + 4x + 7 = -(x+2) + 3\)
=> \(x^2 + 4x + x + 7 - 1\) = 0
=> \(x^2 + 5x + 6 \) = 0
=> \(x^2 + 2x + 3x + 6 \) = 0
=> x(x+2) + 3(x+2) = 0
=> (x+2) (x+3) = 0
=> x = -2 , -3
But the range was x < -2, so x = -3 is the solution in this case
=> x = -3, -2, -1
=> Product of the solutions = -3*-2*-1 = -6
So, Answer will be A
Hope it helps!
Watch the following video to learn How to Solve Absolute Value Problems
