SOLUTION: Hi can someone help me with this question.The question is y=ax^2+bx+c, (-1,0) (3,0) and (2,6) find the equation of the parabola. Can I have step by step process of solving the

Using the parabolic equation, y = ax2 + bx + c, the 1st, (- 1, 0), 2nd, and 3rd points, you do get the following equations:
                             a -  b + c = 0 ------ eq (i)
                            9a + 3b + c = 0 ------ eq (ii)
                            4a + 2b + c = 6 ------ eq (iii)
                            Great job!!


                                8a + 4b = 0 ------ Subtracting eq (i) from eq (ii)                            
                                2a +  b = 0 ------ Factoring out GCF, 4 in the above equation ------ eq (iv) 
                                5a +  b = - 6 ---- Subtracting eq (iii) from eq (ii) ------ eq (v)                   
                                     3a = - 6 ---- Subtracting eq (iv) from eq (v)
                                     

                             2(- 2) + b = 0 ------ Substituting - 2 for a in eq (iv)
                               - 4 + b = 0
                                     

                              - 2 - 4 + c = 0 ------ Substituting - 2 for a, and 4 for b in eq (i)
                                - 6 + c = 0
                                     
With a being - 2, b: 4, and c: 6, the parabolic equation is: 

You can start as you started and continue as other tutors showed you.

This approach and method are general and they work in all cases.

It leads to the system of three linear equations for three unknown coefficients of the quadratic function.



But your case is SPECIAL, and it admits MUCH SIMPLER solution.

Why it is special ? - Because the values  x= -1 and  x= 3 are the ZEROES of the parabola (!)


(it is because the corresponding y-values of the first two points  are zero).



THEREFORE, you can write the quadratic function as 


      y = a*(x-(-1))*(x-3) = a*(x+1)*(x-3)     (1)


with only ONE (!) unknown coefficient "a".



Then we will found this unknown coefficient "a" using the third given point and substituting  x= 2,  y= 6  into the formula (1)

    6 = a*(2+1)*(2-3),   or   6 = -a*3,   a =  = -2.


So, your quadratic function is  y = -2*(x+1)*(x-3),  and you determined it WITHOUT SOLVING ANY EQUATION (i.e. practically MENTALLY).



Thus you know now two ways solving this problem

    - one stupid method of brute force, which works always, but requires solving a system of three equations;

    - and the other SMART method, allowing you solve the problem easy and MENTALLY in special cases.



I will not explain you that a good student should know BOTH methods 
and also should be able to recognize which method to apply for each concrete problem.