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]$x=-6$ or $x=-3$[/tex]
D. no solution
Answer (2)
The solution to the equation x 2 + 49 = x + 5 is x = 5 12 . This was found by squaring both sides, simplifying the equation, isolating x , and verifying the solution. The correct answer is A: 5 12 .
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Square both sides of the equation to eliminate the square root: ( x 2 + 49 ) 2 = ( x + 5 ) 2 .
Expand and simplify the equation: x 2 + 49 = x 2 + 10 x + 25 becomes 49 = 10 x + 25 .
Solve for x : 10 x = 24 , so x = 5 12 .
Verify the solution in the original equation, which confirms that x = 5 12 is a valid solution: 5 12 .
Explanation
Problem Analysis We are given the equation x 2 + 49 = x + 5 and asked to find its solution.
Squaring Both Sides To solve this 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: x 2 + 49 = x 2 + 10 x + 25
Simplifying the Equation Subtracting x 2 from both sides simplifies the equation to: 49 = 10 x + 25
Solving for x Now, we isolate x :
10 x = 49 − 25 10 x = 24 x = 10 24 = 5 12
Checking the Solution We need to check if this solution is valid by substituting x = 5 12 back into the original equation: ( 5 12 ) 2 + 49 = 5 12 + 5 25 144 + 25 1225 = 5 12 + 5 25 25 1369 = 5 37 5 37 = 5 37
Conclusion Since the equation holds true, the solution is valid.
Examples
When designing suspension bridges, engineers use equations involving square roots to calculate cable tension and bridge stability. Solving such equations ensures the bridge can withstand various loads and environmental conditions. For example, if x represents a load factor, the equation x 2 + 49 = x + 5 might model a simplified relationship between the load and the tension in the cables. Finding the correct x ensures the bridge's safety and longevity.
