Given that the quadratic function f (x) satisfies the conditions f (0) = 1, f (x + 1) - f (x) = 2x, the number of solutions of F (| x |) = a, a belonging to R is discussed

Given that the quadratic function f (x) satisfies the conditions f (0) = 1, f (x + 1) - f (x) = 2x, the number of solutions of F (| x |) = a, a belonging to R is discussed

Because f (x) is a quadratic function, Let f (x) = ax ^ 2 + BX + C, so f (x + 1) = a (x + 1) ^ 2 + B (x + 1) + C
So f (x + 1) - f (x) = 2aX + A + B = 2x, so a = 1b = - 1, because f (0) = 1, so C = 1, so f (x) = x ^ 2-x + 1, f (| x |) = | x | ^ 2 - | x | + 1, f (| x |) is even function, draw function image, let y = a
The number of intersections of two images is the number of solutions
When A1, the solution is two
A = 1, 3 solutions
3/4