Select whether the equation has a solution or not.
[tex]2 \sqrt{5 x-4}-3=-11[/tex]
Answer (2)
Isolate the square root term: 2 5 x − 4 = − 11 + 3 .
Simplify: 2 5 x − 4 = − 8 .
Divide by 2: 5 x − 4 = − 4 .
Since the square root of a real number cannot be negative, the equation has no solution: no roots .
Explanation
Problem Analysis We are given the equation 2 5 x − 4 − 3 = − 11 and we want to determine if it has a real solution.
Isolating the Square Root First, we isolate the square root term by adding 3 to both sides of the equation: 2 5 x − 4 = − 11 + 3
Simplifying Simplifying the right side, we get: 2 5 x − 4 = − 8
Dividing by 2 Next, we divide both sides by 2: 5 x − 4 = − 4
Conclusion Since the square root of a real number cannot be negative, there is no real number x that satisfies the equation 5 x − 4 = − 4 . Therefore, the given equation has no real solution.
Examples
Consider a scenario where you are designing a bridge and need to calculate the length of a support cable. The equation 2 5 x − 4 − 3 = − 11 might represent a simplified model of the cable's tension and length. If the equation has no real solution, it indicates that the design parameters are not feasible and need to be adjusted to ensure the bridge's stability. Understanding whether equations have real solutions is crucial in engineering to ensure designs are physically possible and safe.
The equation 2 5 x − 4 − 3 = − 11 has no real solution because it leads to a square root being equal to a negative number, which is not possible. Thus, the conclusion is that there are no roots for this equation.
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