Question

Solve the equation. Write all proposed solutions. If a solution is extraneous, so indicate.
Square root of 3t +31 minus t =9

Answer 1

\[\sqrt{3t+31}-t=9\]

Answer 2

Hi. Add \(t\) to both sides then square both sides.

Answer 3

yeah did that but got \[t ^{2}+15+51\]

Answer 4

\[ \sqrt{3t + 31} - t = 9 \\ \sqrt{3t + 31} = t + 9 \\ 3t + 31 = (t + 9)^2 \\ 3t + 31 = t^2 + 18t + 81 \\ t^2 +15t + 50 = 0 \\ (t +5)(t + 10) = 0 \\ t_1 = -10 \\ t_2 = -5 \]
You have to check these solutions because you squared both sides which is not equivalent operation. So
\[ \sqrt{3 \cdot (-10) + 31} - (-10) = \sqrt{-30+31} + 10 = \sqrt{1} + 10 = 11 \neq 9\]
\[ \sqrt{3 \cdot (-5) + 31} - (-5) = \sqrt{-15 + 31} +5 = \sqrt{16} + 5 = 4 + 5 = 9\]
Hence \(-5\) is only solution of this equation.