For what value(s) of t does the equality ⟨t^3−12t,0.25t^2+4⟩=⟨0,7⟩ hold true?
t=
The notation <....,....> usually represents vectors, so in this case
t^3 - 12t = 0 and .25t^2 + 4 = 7
from the first:
t(t^2 - 12) = 0
t = 0 or t = ±2√3
checking in the 2nd:
if t = 0
LS = 0 ≠ RS
if t = ± 2√3
LS = .25(12) + 4 = 7 = RS
so t = ± 2√3
To find the value(s) of t that satisfies the equation ⟨t^3−12t,0.25t^2+4⟩=⟨0,7⟩, we need to equate the corresponding components of both vectors.
For the first component:
t^3 - 12t = 0
To solve this equation, we can factor out a t:
t(t^2 - 12) = 0
Setting each factor equal to zero gives us two possibilities:
t = 0 and t^2 - 12 = 0
For t = 0, the equation is satisfied.
For t^2 - 12 = 0, we solve for t:
t^2 = 12
t = ±√12
So the possible values for t are t = 0, t = √12, and t = -√12.