CHAPTER 07.05: GAUSS QUADRATURE RULE: Complete Derivation of Two Point Gaussian Quadrature Rule: Part 3 of 3   So once you have proven that x2 equal to minus x1 is acceptable, let's go ahead and see that what does that lead us to? So if x2 is equal to minus x1 is acceptable, let's go ahead and see what happens to the four equations which we had in the beginning. We had the first equation as c1 plus c2 equal to 2, nothing changes there. The next equation was c1 x1, plus c2 x2 is equal to 0, so since x2 is equal to minus x1, so I'll get plus c2 times minus x1 equal to 0. So I'll get c1 minus c2, times x1 equal to 0, so that's what I'm going to get from the second equation.  So this is first equation, and that's the second equation.  Now, let me go ahead and see what happens from the . . . from this equation, I'm getting c1 is equal to c2.  Or, let's look at the third equation, third equation is c1 x1 squared, plus c2 x2 squared is equal to 2/3. And from here, I can say that c1 is equal to c2, because x1 equal to 0 is not a choice, which we just found out.  So c1 is equal to c2 is a choice, so since c1 is equal to c2 is a choice, from here I'll get that c1 and c2 is equal to . . . c1 is equal to c2 is equal to 1, that's what I'm going to get, because you can see that, since x1 equal to 0 is not a choice, as proven earlier, then c1 will be equal to c2, and since c1 is equal to c2 from here, from the first equation that c1 plus c2 is equal to 2, that means that each of them has to be 1 and 1, because they have to be the same, so you have already shown that what the values of c1 and c2 are.  Now, once we have shown that, there is c1 x1 squared, plus c2 x1 squared equal to 2/3, because x2 is equal to minus x1, so this will become x1 squared, and since this is 1 and this is 1, we get 2 x1 squared equal to 2/3, or we get x1 equal to square root of 1 divided by 3, that's what we're going to get for x1, and since x2 is equal to minus x1, that will be -1 divided by square root of 3.  So what you are basically shown is that what these different possibilities are for . . . or what the values of x1, x2, and x3 are, so if I formulate them . . . or bring them together, c1 is 1 c2 is 1, x1 is 1 divided by square root of 3, and x2 is -1 divided by square root of 3. So those are the four numbers which are an acceptable solution to the two-point Gauss quadrature rule formula there. And that's the end of this segment.