Select whether the equation has a solution or not.
[tex]$\sqrt{4 x+5}-\sqrt{x-1}=\sqrt{x+4}$[/tex]
A. no roots
B. roots
Answer (2)
Isolate a square root in the equation.
Square both sides and simplify the equation.
Repeat the process to eliminate all square roots.
Solve the resulting equation for x and verify the solution.
The equation has roots: roo t s
Explanation
Problem Analysis We are given the equation 4 x + 5 − x − 1 = x + 4 and asked to determine if it has a solution.
Isolating a Square Root First, let's isolate one of the square roots: 4 x + 5 = x + 4 + x − 1
Squaring Both Sides Next, we square both sides of the equation: ( 4 x + 5 ) 2 = ( x + 4 + x − 1 ) 2
Expanding the Equation Expanding both sides, we get: 4 x + 5 = ( x + 4 ) + 2 ( x + 4 ) ( x − 1 ) + ( x − 1 )
Simplifying Simplifying, we have: 4 x + 5 = 2 x + 3 + 2 x 2 + 3 x − 4
Isolating the Square Root Term Isolating the remaining square root term: 2 x + 2 = 2 x 2 + 3 x − 4
Dividing by 2 Dividing both sides by 2: x + 1 = x 2 + 3 x − 4
Squaring Again Squaring both sides again: ( x + 1 ) 2 = x 2 + 3 x − 4
Solving for x Expanding and simplifying: x 2 + 2 x + 1 = x 2 + 3 x − 4 , which simplifies to x = 5
Checking the Solution Now, we need to check if x = 5 is a valid solution by substituting it back into the original equation: 4 ( 5 ) + 5 − 5 − 1 = 5 + 4
Verification Simplifying: 25 − 4 = 9 , which gives 5 − 2 = 3 , so 3 = 3 . Thus, x = 5 is a valid solution.
Conclusion Since we found a valid solution, the equation has roots.
Examples
When designing a bridge, engineers use equations involving square roots to calculate the tension and compression forces in the supporting cables and structures. Solving these equations ensures the bridge's stability and safety under various loads. Similarly, in physics, equations with square roots are used to determine the velocity of an object or the time it takes for an object to fall under gravity. These calculations are crucial for accurate predictions and safe designs.
The equation 4 x + 5 − x − 1 = x + 4 has a solution, as we found that x = 5 satisfies the equation when checked back. Hence, the answer is B. roots .
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