Answer:
See below
Step-by-step explanation:
We use the identity A + B = -b/a where A and B are the roots and a and b are the coefficients in ax^2 + bx + c = 0.
So we have
A - B = 1 (given) and
A + B = -b/a
Adding these 2 equations:-
2A = 1 - b/a = (a - b)/a
A = (a - b) / 2a (answer)
Now A - B = 1
B = A - 1 = (a-b)/2a - 1
= - (a + b) / 2a. (answer)
Now the last part:-
AB = (a - b)/2a * - (a + b)/2a
= - (a^2 - b^2) / 4a^2
We now use the identity AB = c/a
so -(a^2 - b^2) / 4a^2 = c/a
Cross multiply:-
-a(a^2 - b^2) = 4a^2c
-a^3 + ab^2 = 4a^2 c
ab^2 = a^3 + 4a^2c
divide through by a:-
b^2 = a^2 + 4ac
b^2 = a( a + 4c) which is what we require.
