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 (2)
Isolate the square root term: x + 8 = x + 6 .
Square both sides: x + 8 = ( x + 6 ) 2 , which simplifies to x 2 + 11 x + 28 = 0 .
Factor the quadratic equation: ( x + 4 ) ( x + 7 ) = 0 , yielding potential solutions x = − 4 and x = − 7 .
Check for extraneous solutions: x = − 4 is a valid solution, but x = − 7 is not. Thus, x = − 4 .
Explanation
Understanding the Problem We are given the equation x + 8 − 6 = x . We need to find the solutions of this equation from the given options.
Isolating the Square Root Isolate the square root: x + 8 = x + 6 .
Squaring Both Sides Square both sides of the equation: ( x + 8 ) 2 = ( x + 6 ) 2 .
Simplifying the Equation Simplify the equation: x + 8 = x 2 + 12 x + 36 .
Rearranging into Quadratic Form Rearrange the equation into a quadratic equation: x 2 + 11 x + 28 = 0 .
Solving the Quadratic Equation Solve the quadratic equation for x . This can be done by factoring or using the quadratic formula. Factoring: ( x + 4 ) ( x + 7 ) = 0 , so x = − 4 or x = − 7 .
Checking for Extraneous Solutions Check the solutions in the original equation to eliminate extraneous solutions. Check x = − 4 : − 4 + 8 − 6 = − 4 ⇒ 4 − 6 = − 4 ⇒ 2 − 6 = − 4 ⇒ − 4 = − 4 . So, x = − 4 is a solution. Check x = − 7 : − 7 + 8 − 6 = − 7 ⇒ 1 − 6 = − 7 ⇒ 1 − 6 = − 7 ⇒ − 5 = − 7 . So, x = − 7 is not a solution.
Final Solution Therefore, the only solution is x = − 4 .
Examples
When designing a suspension bridge, engineers use equations involving square roots to calculate the tension in the cables. Solving these equations helps ensure the bridge's stability and safety. Similarly, in physics, determining the velocity of an object under constant acceleration involves solving equations with square roots, which is crucial for predicting the object's motion and preventing collisions.
The only solution to the equation x + 8 − 6 = x is x = − 4 , which has been checked and validated in the original equation. Therefore, the correct answer from the choices provided is C. x = − 4 .
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