Use the form y=ax2 + bx + c, for quadratic equations Write the equation for the following relation. D = {(x, y): (3, 9), (5, 25), (-2, 4), . . .}

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Y=ax^2 + bx + c
1.  9 = a*3^2 + b*3 +c
2.  25 = a*5^2 + b*5 +c
3.  4 = a*(-2)^2 + b*(-2) + c
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Let's make the equations more simple:
1.   9 = 9a + 3b + c
2.   25 = 25a + 5b + c
3.   4 = 4a -2b + c
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Let's reduce the number of equations to 2.  We'll do it as follows: 
From equation no. 3  we see that c = -4a +2b +4    Now we shall put that value instead of c in the other 2 equations:
1.  9 = 9a + 3b -4a +2b +4  
     that will make:  5 = 5a +5b   
     or 1 = a +b     (getting simple, hey ?)
2.  25 = 25a + 5b -4a +2b +4
     that will make:  21 = 21a + 7B
     or (divide by 7)    3 = 3a + b
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Now:  from 1 = a + b   we get   a = 1-b
 Let us install it in equation no. 2 to get:
3 = 3 (1 - b) + b
3 = 3 - 3b +b
0 = -2b     that means b = 0
Since a = 1 -b and b=0 that means a= 1
Now go back  c = -4a +2b +4 or 
c = -4 + 4    
c = 0

So the equation looks as follows:  Y = X^2

 

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