Solve the radical equation. Check for extraneous solutions. [tex]$\sqrt{t}+7=4$[/tex]
A. [tex]$t=-3$[/tex]
B. No solution
C. [tex]$t =9$[/tex]
D. [tex]$t=-9$[/tex]
Answer (2)
Isolate the radical: t = − 3 .
Square both sides: t = 9 .
Check for extraneous solutions: 9 + 7 = 10 = 4 .
The equation has no solution: No solution .
Explanation
Understanding the Problem We are given the radical equation t + 7 = 4 and asked to solve for t , checking for any extraneous solutions. Extraneous solutions can arise when we square both sides of an equation, so it's crucial to verify our solutions in the original equation.
Isolating the Radical First, we isolate the radical term by subtracting 7 from both sides of the equation: t + 7 − 7 = 4 − 7 t = − 3
Squaring Both Sides Now, we square both sides of the equation to eliminate the square root: ( t ) 2 = ( − 3 ) 2 t = 9
Checking for Extraneous Solutions Next, we must check if t = 9 is an extraneous solution by substituting it back into the original equation: 9 + 7 = 4 3 + 7 = 4 10 = 4 Since 10 = 4 , t = 9 is an extraneous solution.
Final Answer Since the square root of a real number cannot be negative, the equation t = − 3 has no real solution. Therefore, the original equation t + 7 = 4 has no solution.
Examples
Radical equations appear in various fields, such as physics and engineering, when dealing with relationships involving square roots or other radicals. For example, when calculating the period of a pendulum, the formula involves a square root. Solving radical equations allows us to determine unknown variables within these relationships. Understanding how to solve these equations and check for extraneous solutions ensures accurate and reliable results in practical applications.
The equation t + 7 = 4 leads to t = − 3 , which is impossible since square roots cannot be negative. Thus, the original equation has no solution, and the correct option is B: No solution.
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