How do you find the quadratic coefficient a, b, and c, given some values of x and y?

For example: Suppose we have f(0.1524)=0.9961 f(0.8258)=0.0782 f(0.5383)=0.4427 and given the quadratic equation: $y = ax^2 + bx + c$, then can you find values of x and y?

Is there a formula to find these a, b, and c? when I think about the general formula, but in this case y=0. So, it does't work. When I think about substitute by the formula: $y = ax^2 + bx + c$, I cannot get go long; because we have three unknown in one equation. So, what you suggest as next step.

Thank you!

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    $\begingroup$ You get three linear equations for three unknowns. $\endgroup$ – H. Gutsche Jan 17 '19 at 15:47
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    $\begingroup$ Just solve a system of equations for $a$, $b$, and $c$ by plugging in $x$ and $y$ in the equation $y = ax^2+bx+c$. $\endgroup$ – KM101 Jan 17 '19 at 15:47
  • $\begingroup$ Thank you both H.Gutsche and KM101. It is very clear now. $\endgroup$ – user777 Jan 17 '19 at 15:57

If you are given $3$ points with distinct $x$ values, by substituting those values inside $y=ax^2+bx+c$, we can obtain $3$ linear equations in terms of $a, b$, and $c$.

Hence, the problem reduces to the problem of solving a linear system of equations, which can be solved, for example, by Gaussian elimination.

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