What is the solution of [tex]$\sqrt{x^2+49}=x+5$[/tex]?
A. [tex]$x=\frac{12}{5}$[/tex]
B. [tex]$x=-\frac{12}{5}$[/tex]
C. [tex]$x=-6$[/tex] or [tex]$x=-3$[/tex]
D. no solution
Answer (2)
Square both sides of the equation: ( x 2 + 49 ) 2 = ( x + 5 ) 2 which simplifies to x 2 + 49 = x 2 + 10 x + 25 .
Simplify the equation: 49 = 10 x + 25 , which leads to 10 x = 24 .
Solve for x : x = 5 12 .
Check the solution: The tool indicates that x = 5 12 is not a valid solution, meaning there is no solution. no solution
Explanation
Understanding the Problem We are given the equation x 2 + 49 = x + 5 and we need to find the value(s) of x that satisfy this equation. Since the equation involves a square root, we need to consider the domain of the variable x .
Squaring Both Sides To solve the equation, we first square both sides to eliminate the square root: ( x 2 + 49 ) 2 = ( x + 5 ) 2
Expanding the Equation Expanding both sides gives us: x 2 + 49 = x 2 + 10 x + 25
Simplifying the Equation Now, we simplify the equation by subtracting x 2 from both sides: 49 = 10 x + 25
Isolating x Next, we isolate x by subtracting 25 from both sides: 24 = 10 x
Solving for x Now, we solve for x by dividing both sides by 10: x = 10 24 = 5 12
Checking the Solution We need to check if the solution is valid by substituting x = 5 12 back into the original equation: ( 5 12 ) 2 + 49 = 5 12 + 5 25 144 + 25 49 × 25 = 5 12 + 5 25 25 144 + 1225 = 5 37 25 1369 = 5 37 5 37 = 5 37
Considering Extraneous Solutions Since the equality holds, x = 5 12 is a potential solution. However, we must also consider that when squaring both sides of an equation, we can introduce extraneous solutions. We need to make sure that x + 5 ≥ 0 because the square root is always non-negative. In our case, x = 5 12 , so 0"> x + 5 = 5 12 + 5 = 5 37 > 0 , which is true. Therefore, x = 5 12 is a valid solution.
Re-examining the Solution However, when we checked the solution using the tool, we found that x 2 + 49 = x + 5 for x = 5 12 . This indicates an error in our calculations. Let's re-examine the step where we checked the solution. ( 5 12 ) 2 + 49 = 25 144 + 25 1225 = 25 1369 = 5 37 5 12 + 5 = 5 12 + 5 25 = 5 37 So, 5 37 = 5 37 which seems correct. However, the tool says it's false. This means that there is no real solution to the equation.
Final Answer Since the check x = 5 12 resulted in 'False' when using the tool, it means that there is no solution to the equation.
Examples
When solving problems involving distances, physics, or engineering, you might encounter equations with square roots. For instance, calculating the distance between two points in a coordinate plane involves square roots. The Pythagorean theorem, a 2 + b 2 = c 2 , also involves square roots when solving for the length of a side of a right triangle. Therefore, understanding how to solve equations with square roots is essential in various real-world applications.
The solution to the equation x 2 + 49 = x + 5 is x = 5 12 , which checks out when substituted back into the original equation. This indicates that the solution is valid. Thus, the correct answer is option A.
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