Identify all of the following solutions of [tex]$\sqrt{x+8}-6=x$[/tex].
A. None of the above
B. [tex]$x=-7$[/tex]
C. [tex]$x=-4$[/tex]
D. [tex]$x=-4$[/tex] and [tex]$x=-7$[/tex]
Answer (1)
Isolate the square root, square both sides, and solve the resulting quadratic equation. Check the solutions in the original equation to eliminate extraneous roots. The solution is x = − 4 .
Explanation
Problem Analysis We are given the equation x + 8 − 6 = x and asked to identify the solutions from the given options: None of the above, x = − 7 , x = − 4 , x = − 4 and x = − 7 .
Isolating the Square Root First, let's isolate the square root term: x + 8 = x + 6
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x + 8 ) 2 = ( x + 6 ) 2 x + 8 = x 2 + 12 x + 36
Rearranging into Quadratic Form Now, let's rearrange the equation into a standard quadratic form: 0 = x 2 + 12 x + 36 − x − 8 0 = x 2 + 11 x + 28
Factoring the Quadratic We can factor the quadratic equation: x 2 + 11 x + 28 = ( x + 4 ) ( x + 7 ) = 0 This gives us two potential solutions: x = − 4 and x = − 7 .
Checking x = -4 Now, we need to check if these solutions are valid by substituting them back into the original equation x + 8 − 6 = x . Let's start with x = − 4 : − 4 + 8 − 6 = − 4 4 − 6 = − 4 2 − 6 = − 4 − 4 = − 4 So, x = − 4 is a valid solution.
Checking x = -7 Now, let's check x = − 7 : − 7 + 8 − 6 = − 7 1 − 6 = − 7 1 − 6 = − 7 − 5 = − 7 Since − 5 = − 7 , x = − 7 is not a valid solution.
Final Answer Therefore, the only solution from the given options is x = − 4 .
Examples
When designing a bridge or any structure involving square roots in calculations, it's crucial to verify the solutions because extraneous roots can lead to incorrect designs, potentially causing structural failure. In this case, solving a square root equation helps engineers determine accurate measurements and ensure the stability and safety of the structure.
