System of Equations Mechanics problem

[Punchlinegirl]

The x-coordinates of two objects moving along the x-axis are given below as a function of time t. $$x_1$$ and $$x_2$$ never have the same value. Calculate the value of $$x_2$$ when the objects are nearest to each other.
$$x_1$$= 23.0t
$$x_2= -28.0 +43.0t-8.0t^2$$

I solved the first equation for t, and then plugged it into the second one to get $$x_2= -28.0 + 43.0( \frac {x_1}{23}) -8.0(\frac {x_1}{23})^2$$

I tried to use the quadratic formula but got a negative number... can someone tell me what I'm doing wrong? Thanks in advance.

 

[lightgrav]

what value did you use for x_2 in your quadratic formula?
it's easier algebra to replace those "x1/23" terms with "t".

This is a relative location question ... what is x2 rel. to x1?
now there's only one equation, one unknown (t).
minimize x2-x1 , by setting t-derivitive =0.

 

[HallsofIvy]

If you mean you completed the square to find the minimum value of the quadratic then, since that quadratic is equal to x₂, you are just finding the minimum value of x₂ itself, not where the distance between x₁ and x₂ is a minimum.

The distance between x₁ and x₂ is |x₁- x₂|- that's what you want to minimize.

It's probably simplest to look at x₁- x₂ and x₂- x₁ separately.

 

[Punchlinegirl]

Ok I got it... thanks for your help