1. Calculate Fitting second degree parabola - Curve fitting using Least square method

X	Y
1	-5
2	-2
3	5
4	16
5	31
6	50
7	73

Solution:
The equation is `y = a + bx + cx^2` and the normal equations are

`sum y = an + b sum x + c sum x^2`

`sum xy = a sum x + b sum x^2 + c sum x^3`

`sum x^2y = a sum x^2 + b sum x^3 + c sum x^4`


The values are calculated using the following table

`x`	`y`	`x^2`	`x^3`	`x^4`	`x*y`	`x^2*y`
1	-5	1	1	1	-5	-5
2	-2	4	8	16	-4	-8
3	5	9	27	81	15	45
4	16	16	64	256	64	256
5	31	25	125	625	155	775
6	50	36	216	1296	300	1800
7	73	49	343	2401	511	3577
---	---	---	---	---	---	---
`sum x=28`	`sum y=168`	`sum x^2=140`	`sum x^3=784`	`sum x^4=4676`	`sum x*y=1036`	`sum x^2*y=6440`

Substituting these values in the normal equations
`7a+28b+140c=168`

`28a+140b+784c=1036`

`140a+784b+4676c=6440`


Solving these 3 equations,
Total Equations are `3`

`7a+28b+140c=168 -> (1)`

`28a+140b+784c=1036 -> (2)`

`140a+784b+4676c=6440 -> (3)`

---

Select the equations `(1)` and `(2)`, and eliminate the variable `a`.

`7a+28b+140c=168`	` xx 4->`		``	`28a`	`+`	`112b`	`+`	`560c`	`=`	`672`	``
		−
`28a+140b+784c=1036`	` xx 1->`		``	`28a`	`+`	`140b`	`+`	`784c`	`=`	`1036`	``
					`-`	`28b`	`-`	`224c`	`=`	`-364`	` -> (4)`

---

Select the equations `(1)` and `(3)`, and eliminate the variable `a`.

`7a+28b+140c=168`	` xx 20->`		``	`140a`	`+`	`560b`	`+`	`2800c`	`=`	`3360`	``
		−
`140a+784b+4676c=6440`	` xx 1->`		``	`140a`	`+`	`784b`	`+`	`4676c`	`=`	`6440`	``
					`-`	`224b`	`-`	`1876c`	`=`	`-3080`	` -> (5)`

---

Select the equations `(4)` and `(5)`, and eliminate the variable `b`.

`-28b-224c=-364`	` xx 8->`				`-`	`224b`	`-`	`1792c`	`=`	`-2912`	``
		−
`-224b-1876c=-3080`	` xx 1->`				`-`	`224b`	`-`	`1876c`	`=`	`-3080`	``
							``	`84c`	`=`	`168`	` -> (6)`

---

Now use back substitution method
From (6)
`84c=168`

`=>c=(168)/(84)=2`

From (4)
`-28b-224c=-364`

`=>-28b-224(2)=-364`

`=>-28b-448=-364`

`=>-28b=-364+448=84`

`=>b=(84)/(-28)=-3`

From (1)
`7a+28b+140c=168`

`=>7a+28(-3)+140(2)=168`

`=>7a+196=168`

`=>7a=168-196=-28`

`=>a=(-28)/(7)=-4`

Solution using Elimination method.
`a=-4,b=-3,c=2`

Now substituting this values in the equation is `y = a + bx + cx^2`, we get

`y = -4 -3x+2x^2`