Question

The yearly profit of a company in thousands of dollars can be modeled by P(t) = t^4 − 5t^2 + 4, where t is the number of years since 2000. Factor to find the years in which the profit was 0.

Answer 1

when the profit is 0; then P(t)=0
You have this equation:
t⁴-5t²+4=0

Then, you have to do the following change:
t²=z

you have this other equation:
z²-5t+4=0

Now, you can solve this square equation:
z=[5⁺₋√(25-16)]/2=(5⁺₋√9)/2=
you have two solutions for z.
z₁=(5-3)/2=1
z₂=(5+3)/2=4

Now; if z=1; then
t²=z
t²=1
t=⁺₋√1

We have two solutions:
t=-1 this is not a valid solution:
t=1;

Then in 2001  (2000+t=2000+1=2001) the profit was 0.

If z=4, then:
t²=z
t²=1
t=⁺₋√4

We have two solutions:
t=-2 this is not a valid solution:
t=2;

Then in 2002  (2000+t=2000+2=2002) the profit was 0.


Answer: the yeares in which the profict was 0, were 2001 and 2002.