Find the quadratic function that is the best fit for $f(x)$ defined by the table below. $\newline$ \begin{tabular}{|c|c|c|c|c|c|c|} $\newline$ \hline $\mathbf{x}$ & $0$ & $2$ & $4$ & $6$ & $8$ & $10$ \\ $\newline$ \hline $\mathbf{f ( x )}$ & $0$ & $397$ & $1603$ & $3605$ & $6402$ & $9999$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ The quadratic function is $y=\square x^{2}+\square x+(\square)$ . $\newline$ (Type an integer or decimal rounded to two decimal places as needed.)

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Math Problems

Algebra 2

Quadratic equation with complex roots

Full solution

Q. Find the quadratic function that is the best fit for

f(x)

defined by the table below.

\newline

\begin{tabular}{|c|c|c|c|c|c|c|}

\newline

\hline

\mathbf{x}

0

2

4

6

8

10

\newline

\hline

\mathbf{f ( x )}

0

397

1603

3605

6402

9999

\newline

\hline

\newline

\end{tabular}

\newline

The quadratic function is

y=\square x^{2}+\square x+(\square)

\newline

(Type an integer or decimal rounded to two decimal places as needed.)

Form Quadratic Function: To find the best fit quadratic function, we'll use the form $y = ax^2 + bx + c$ and solve for $a$ , $b$ , and $c$ using the given points.
Plug in Points: Plug in the points $(0, 0)$ , $(2, 397)$ , $(4, 1603)$ , $(6, 3605)$ , $(8, 6402)$ , and $(10, 9999)$ into the equation $y = ax^2 + bx + c$ .
Solve for Coefficients: For $(0, 0)$ : $0 = a(0)^2 + b(0) + c$ , which simplifies to $c = 0$ .
Set Up Matrix Equation: For $(2, 397)$ : $397 = a(2)^2 + b(2) + c$ , which simplifies to $397 = 4a + 2b + c$ .
Calculate Sums: For $(4, 1603)$ : $1603 = a(4)^2 + b(4) + c$ , which simplifies to $1603 = 16a + 4b + c$ .
Substitute Sums: For $(6, 3605)$ : $3605 = a(6)^2 + b(6) + c$ , which simplifies to $3605 = 36a + 6b + c$ .
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$ Substitute these sums into the normal equations to get a system of equations for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$ Substitute these sums into the normal equations to get a system of equations for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .The system of equations is: $\newline$ $9999 = a(10)^2 + b(10) + c$ $6$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $7$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $8$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$ Substitute these sums into the normal equations to get a system of equations for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .The system of equations is: $\newline$ $9999 = a(10)^2 + b(10) + c$ $6$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $7$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $8$ Since $c = 0$ , the system simplifies to: $\newline$ $9999 = 100a + 10b + c$ $0$ $\newline$ $9999 = 100a + 10b + c$ $1$ $\newline$ $9999 = 100a + 10b + c$ $2$
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$ Substitute these sums into the normal equations to get a system of equations for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .The system of equations is: $\newline$ $9999 = a(10)^2 + b(10) + c$ $6$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $7$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $8$ Since $c = 0$ , the system simplifies to: $\newline$ $9999 = 100a + 10b + c$ $0$ $\newline$ $9999 = 100a + 10b + c$ $1$ $\newline$ $9999 = 100a + 10b + c$ $2$ Solve the system using matrix operations or substitution/elimination methods to find $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .
Solve System of Equations: For $(8, 6402)$ : $6402 = a(8)^2 + b(8) + c$ , which simplifies to $6402 = 64a + 8b + c$ .For $(10, 9999)$ : $9999 = a(10)^2 + b(10) + c$ , which simplifies to $9999 = 100a + 10b + c$ .Now we have a system of equations: $\newline$ $1$ . $c = 0$ $\newline$ $2$ . $397 = 4a + 2b + c$ $\newline$ $3$ . $1603 = 16a + 4b + c$ $\newline$ $4$ . $3605 = 36a + 6b + c$ $\newline$ $5$ . $6402 = 64a + 8b + c$ $\newline$ $6$ . $9999 = 100a + 10b + c$ Since $c = 0$ , we can simplify the equations: $\newline$ $2$ . $6402 = a(8)^2 + b(8) + c$ $3$ $\newline$ $3$ . $6402 = a(8)^2 + b(8) + c$ $4$ $\newline$ $4$ . $6402 = a(8)^2 + b(8) + c$ $5$ $\newline$ $5$ . $6402 = a(8)^2 + b(8) + c$ $6$ $\newline$ $6$ . $6402 = a(8)^2 + b(8) + c$ $7$ Use the method of least squares to solve for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ . This involves setting up a matrix equation and solving for the coefficients.The matrix equation is based on the normal equations of the least squares method, which are derived from the system of equations.The normal equations are: $\newline$ $6402 = 64a + 8b + c$ $0$ $\newline$ $6402 = 64a + 8b + c$ $1$ $\newline$ $6402 = 64a + 8b + c$ $2$ Substitute the sums $6402 = 64a + 8b + c$ $3$ , $6402 = 64a + 8b + c$ $4$ , $6402 = 64a + 8b + c$ $5$ , $6402 = 64a + 8b + c$ $6$ , $6402 = 64a + 8b + c$ $7$ , $6402 = 64a + 8b + c$ $8$ , and $6402 = 64a + 8b + c$ $9$ from the given points into the normal equations.Calculate the sums: $\newline$ $(10, 9999)$ $0$ $\newline$ $(10, 9999)$ $1$ $\newline$ $(10, 9999)$ $2$ $\newline$ $(10, 9999)$ $3$ $\newline$ $(10, 9999)$ $4$ $\newline$ $(10, 9999)$ $5$ $\newline$ $(10, 9999)$ $6$ After calculating the sums, we get: $\newline$ $(10, 9999)$ $7$ $\newline$ $(10, 9999)$ $8$ $\newline$ $(10, 9999)$ $9$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $0$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $1$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $2$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $3$ Substitute these sums into the normal equations to get a system of equations for $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .The system of equations is: $\newline$ $9999 = a(10)^2 + b(10) + c$ $6$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $7$ $\newline$ $9999 = a(10)^2 + b(10) + c$ $8$ Since $c = 0$ , the system simplifies to: $\newline$ $9999 = 100a + 10b + c$ $0$ $\newline$ $9999 = 100a + 10b + c$ $1$ $\newline$ $9999 = 100a + 10b + c$ $2$ Solve the system using matrix operations or substitution/elimination methods to find $6402 = a(8)^2 + b(8) + c$ $8$ and $6402 = a(8)^2 + b(8) + c$ $9$ .After solving, we find that $9999 = 100a + 10b + c$ $5$ , $9999 = 100a + 10b + c$ $6$ , and $c = 0$ .