Mathematics Asked on January 3, 2022

Joel is thinking of a quadratic and Eve is thinking of a quadratic. Both use $x$ as their variable. When they evaluate their quadratics for $x=1$, they get the same number. When they evaluate their quadratics for $x=2$, they both again get the same number. And when they evaluate their quadratics for $x=3$, they *again* both have the same result. Are their quadratics necessarily the same?

If $x=1$ results in $k_1$, $x=2$ in $k_2$ and $x=3$ in $k_3$ then three equations can be made by inputting these values in $ax^2+bx+c=k_i$

$a+b+c=k_1$

$4a+2b+c=k_2$

$9a+3b+c=k_3$

Using these equations we find the quadratic coefficients in terms of $k_i$:

$a=frac{k_1-2k_2+k_3}{2}$

$b=frac{-5k_1+8k_2-3k_3}{2}$

$c=3k_1-3k_2+k_3$

Now I’m stuck on what to do, hitherto I’ve followed hints but the answer still eludes me.

The graph of a quadratic is a parabola, which is defined by three points, or three inputs and outputs. Since the two quadratics have three inputs which give identical outputs, they must be the same quadratic.

You might be wondering how to prove that a parabola is defined by three points or three inputs and outputs. Here's a link with the exact question: Prove that three points define a unique parabola

Hope this helped!

Answered by OlympusHero on January 3, 2022

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