Find the non-extraneous solutions of [tex]$\sqrt{x+7}-4=x+3$[/tex].
A. [tex]$x=-6$[/tex] and [tex]$x=-7$[/tex]
B. [tex]$x=-2$[/tex]
C. [tex]$x=-7$[/tex] and [tex]$x=-2$[/tex]
D. [tex]$x=1$[/tex]
Answer (2)
Isolate the square root: x + 7 = x + 7 .
Square both sides: x + 7 = ( x + 7 ) 2 .
Simplify to a quadratic equation: x 2 + 13 x + 42 = 0 .
Factor and solve: ( x + 6 ) ( x + 7 ) = 0 , so x = − 6 and x = − 7 . Check for extraneous solutions. The final answer is x = − 6 and x = − 7 .
Explanation
Understanding the Problem We are given the equation x + 7 − 4 = x + 3 and asked to find the non-extraneous solutions. This means we need to solve the equation and then check if the solutions we find actually satisfy the original equation. Extraneous solutions can arise when we square both sides of an equation, so checking our solutions is crucial.
Isolating the Square Root First, let's isolate the square root term: x + 7 = x + 3 + 4 x + 7 = x + 7
Squaring Both Sides Now, square both sides of the equation to eliminate the square root: ( x + 7 ) 2 = ( x + 7 ) 2 x + 7 = ( x + 7 ) 2
Expanding and Simplifying Expand the right side and simplify the equation: x + 7 = x 2 + 14 x + 49 0 = x 2 + 13 x + 42
Factoring the Quadratic Factor the quadratic equation: 0 = ( x + 6 ) ( x + 7 )
Finding Potential Solutions Solve for x :
x + 6 = 0 ⇒ x = − 6 x + 7 = 0 ⇒ x = − 7
Checking for Extraneous Solutions Now, we need to check for extraneous solutions by plugging each potential solution back into the original equation x + 7 − 4 = x + 3 .
For x = − 6 :
− 6 + 7 − 4 = − 6 + 3 1 − 4 = − 3 1 − 4 = − 3 − 3 = − 3
This is true, so x = − 6 is a valid solution.
For x = − 7 :
− 7 + 7 − 4 = − 7 + 3 0 − 4 = − 4 0 − 4 = − 4 − 4 = − 4 This is true, so x = − 7 is a valid solution.
Final Answer Both x = − 6 and x = − 7 are valid solutions to the original equation. Therefore, the non-extraneous solutions are x = − 6 and x = − 7 .
Stating the Solutions The solutions are x = − 6 and x = − 7 .
Examples
When designing a bridge, engineers use equations involving square roots to calculate the tension and compression forces acting on different parts of the structure. Finding the solutions to these equations ensures the bridge's stability. However, sometimes the mathematical solutions don't make sense in the real world (like a negative length), so engineers must check for extraneous solutions to ensure their calculations are physically meaningful and safe.
The non-extraneous solutions to the equation x + 7 − 4 = x + 3 are x = − 6 and x = − 7 , both of which satisfy the original equation upon verification. Therefore, the answer is option A, x = − 6 and x = − 7 .
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