Rearranging numbers changes answers?

[Lexadis]

Solved~Rearranging numbers changes answers?

Homework Statement

A car starts from rest and travels with a constant acceleration of $3ms^{-2}$, while a bike which is at a distance of 100m away from the car starts with an initial velocity of $5ms^{-2}$ travels with a constant acceleration of $2ms^{-2}$. The displacement traveled by the bike before being overtaken is $x$. Using equations of motion, find,
(i) the time taken for the car to overtake the bike.
(ii) the distance traveled by the bike (x)
(iii) the distance traveled by the car.

Homework Equations

$s= ut + \frac{1}{2} at^{2}$

The Attempt at a Solution

So I did using the theory that the displacement of the car will be equal to displacement of the bike +100m.
So here it goes:
$ut+ \frac{1}{2}at^{2}+100 = ut+ \frac{1}{2}at^{2}$

$\frac{1}{2}.3.t^{2} + 100 = 5t + \frac{1}{2}.2.t^{2}$

$\frac{3}{2}t^{2}+100=5t+t^{2}$

$100=5t-\frac{1}{2}t^{2}$

$\frac{1}{2}t^{2}-5t+100=0$

$t^{2}-10t+200=0$

It is after this the problem started. I got 2 different answers. Here it goes:

$t^{2}-10t+200=0$

--> Here I arranged it as -20t + 10t:

$t^{2} - 20t + 10t + 200 = 0$

$t(t-20)+10(t+20)=0$

$(t+10)(t-20)(t+20) = 0$

$t+10 = 0 / t^{2}-20^{2} = 0$

$t=-10 / t=20 s / t=-20s$

-->Here I arranged it as +10, -20, and got different answers >_>

$t^{2} + 10t - 20t + 200 = 0$

$t(t+10) - 20(t-10) = 0$

$(t-20) (t^{2}-10^{2}) = 0$

$t=20s / t=10 / t=-10$

The answer is supposed to be 20. So how come I also got t=10s in my second arrangement? Is there some mistake I can't identify? Thank you very much in advance c:

 

[vela]

You need to review your algebra. You can't factor the way you did. Note that you started with a quadratic expression, but after factoring, you ended up with a cubic expression. Obviously, you did something you're not allowed to do.

To be specific, $t(t-20) + 10(t+20)$ does not equal $(t+10)(t-20)(t+20)$. You made a similar mistake in the second approach as well.

 

[Lexadis]

I figured out what my mistake was ^^;
I should have had:
$s=ut+\frac{1}{2}at^{2}-100 = 5t+t^{2}$
instead of a $+100$ because displacement of the car minus100m should be equal to the displacement of the bike.
Thank you!

 

[BiGyElLoWhAt]

ummm... to reiterate what vela said:

firstly you go from
t(t−20)+10(t+20)=0
to
(t+10)(t−20)(t+20)=0

which is nowhere near correct:
(t+10)(t−20)(t+20)=(t+10)(t^2-400) = t^3 +10t^2 -400t - 4000
definitely not your origional equation.
Same thing with your second attempt. Bad algebra is bad.

 

[HalfLostAndSearching]

Yes, rearranging numbers in equations can change the answers. This is because the equations are based on mathematical relationships between the different variables, and changing the order of the terms can change the way those relationships are expressed. In this particular case, rearranging the terms in the equation t^2 - 10t + 200 = 0 can lead to incorrect solutions because it changes the relationship between the variables t and t^2. It is important to use the correct equation and to solve it in a consistent manner in order to obtain the correct answer. In this case, using the quadratic formula would be a more reliable method for solving the equation.