The normal equations for a second degree parabola y = ax2 + bx + c are Σy = aΣx2 + bΣx + nc, Σxy = aΣx3 + bΣx2 + cΣx and Σx2y = aΣx4 + bΣx3 + cΣx2.. Is it true or false?a)Trueb)FalseCorrect answer is option 'A'. Can you explain this answer?

Question

The normal equations for a second degree parabola y = ax2 + bx + c are &Sigma;y = a&Sigma;x2 + b&Sigma;x + nc, &Sigma;xy = a&Sigma;x3 + b&Sigma;x2 + c&Sigma;x and &Sigma;x2y = a&Sigma;x4 + b&Sigma;x3 + c&Sigma;x2.. Is it true or false?a)Trueb)FalseCorrect answer is option 'A'. Can you explain this answer?

Answer

The normal equations for a second degree parabola y = ax^2 + bx + c are:

1) ∂E/∂a = 0, where E is the error function defined as the sum of squared differences between the actual y-values and the predicted y-values for a given set of x-values. This equation gives the derivative of E with respect to a, and setting it to 0 gives the value of a that minimizes the error.

2) ∂E/∂b = 0, where ∂E/∂b is the derivative of E with respect to b, and setting it to 0 gives the value of b that minimizes the error.

3) ∂E/∂c = 0, where ∂E/∂c is the derivative of E with respect to c, and setting it to 0 gives the value of c that minimizes the error.

These equations can be solved simultaneously to find the values of a, b, and c that best fit the given data points.

Answer

The second order parabola is given by
y = ax² + bx +c.
Let the sum of residues be E.
E = Σd_i² = Σ(y_i – (ax_i² + bxi +c))².
Here

Solving these three equations, we get the normal equations as
Σy = aΣx² + bΣx + nc, Σxy = aΣx³ + bΣx² + cΣx and Σx²y = aΣx⁴ + bΣx³ +cΣx².

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