In solving Real World Quadratic Equations, why does change value, and when do i have to change value?

Question

When solving for the equation I entered 32, not -32. When I watched the video paired with this course it did not explain in what cases to do this. All help is appreciated :D

Answer 1

The reason for the change of $32$ to $-32$ (as well as the other changes in sign, you seemed to have missed those) is minor: oftentimes, people like to have their quadratics with leading coefficient (that is, that of the $x^2$ term, or $t^2$ in your case) as positive. This is achieved by multiplying through by negative one, in this case.

$$32 - 2t - 5t^2 = 0 \implies -1(32 - 2t - 5t^2) = -1(0) \implies 5t^2 + 2t - 32 = 0$$

Visually, this is like flipping the quadratic about the $x$ axis (or $t$ in this case), which preserves the roots.

Of course, one then has to ask - since you achieve the same answer in solving for the roots this way, what exactly is the point? One could even argue that not multiplying through by the negative is better, since you can more readily see the behavior of the original function.

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In short, the reason for the sign change is the desire to have a positive coefficient on the $t^2$ term and thus warranted multiplying through by $-1$.

But at the same time, in solving for the roots, you get the same answer even if you don't multiply through by negative one, warranting this change unnecessary and arbitrary. I honestly can't really see the point.

Answer 2

In the problem that you presented, the explanation is subtracting the terms in the first equation from both sides for convenience. This is not necessary, and the answer can be obtained without doing so. However, in that case, you will also have -5 instead of 5 (and -2 instead of 2) as well.