Solve for t:
Radical(t-108) - Radical(t+186) = 6
The only possible root is t = ?. It is an extraneous root.
To solve for t in the equation Radical(t-108) - Radical(t+186) = 6, we need to isolate the radical terms on one side of the equation and then square both sides.
1. Start by adding Radical(t+186) to both sides of the equation:
Radical(t-108) = Radical(t+186) + 6
2. Next, square both sides of the equation to eliminate the radicals:
(Radical(t-108))^2 = (Radical(t+186) + 6)^2
Squaring both sides gets rid of the radicals, but it also introduces extraneous solutions. So, it's important to check for extraneous roots at the end.
3. Simplify the left side by taking the square:
t - 108 = (Radical(t+186) + 6)^2
4. Expand the right side by using the square of a binomial formula:
t - 108 = t + 186 + 12(Radical(t+186)) + 36
5. Combine like terms and rearrange the equation:
t - t - 306 = 12(Radical(t+186))
-306 = 12(Radical(t+186))
6. Divide both sides by 12 to isolate the radical term:
-25.5 = Radical(t+186)
7. Now, square both sides again to get rid of the radical:
(-25.5)^2 = (Radical(t+186))^2
650.25 = t + 186
8. Solve for t by moving the constants to the other side:
t = 650.25 - 186
t ≈ 464.25
Now, we have a potential solution t ≈ 464.25. However, we need to check if it is an extraneous root to ensure it satisfies the original equation.
9. Substitute t ≈ 464.25 back into the original equation:
Radical(464.25 - 108) - Radical(464.25 + 186) = 6
After evaluating both radicals, the equation becomes:
6 - Radical(650.25) = 6
Simplifying further:
6 - 25.5 = 6
-19.5 ≠ 6
Since -19.5 is not equal to 6, the potential solution t ≈ 464.25 is an extraneous root. Therefore, there is no valid solution to the original equation.