The number of solutions of equation | x ^ 2-2x | = a ^ 2 + 1 (a ∈ R +) A1 B2 C3 D4

The number of solutions of equation | x ^ 2-2x | = a ^ 2 + 1 (a ∈ R +) A1 B2 C3 D4

|X ^ 2-2x | = a ^ 2 + 1, so x ^ 2-2x = a ^ 2 + 1 or x ^ 2-2x = - A ^ 2-1
That is: x ^ 2-2x-a ^ 2-1 = 0 or x ^ 2-2x + A ^ 2 + 1 = 0
Δ1=4(a^2+2)>0,Δ2=-4a^2

Find the number of different solutions of the equation | X - | 2x + 1 | = 3

|X - | 2x + 1 | = 3, when x = - 12, the original equation is transformed into | x | = 3, no solution; when x ＞ - 12, the original equation is transformed into | 1 + X | = 3, the solution is x = 2 or x = - 4 (rounding off). When x ＜ - 12, the original equation can be transformed into | x + (2x + 1) | = 3, that is | 3x + 1 | = 3, | 3x + 1 = ± 3, the solution is x = 23 (rounding off) or x = - 43

There are several solutions to the equation a ^ x = - x ^ 2 + 2x + a (a > 0, and a ≠ 0) of X

a>0
Y = - x ^ 2 + 2x + A, vertex (1, a + 1)
"The vertex of y = ax & sup2; + BX + C is (- B / 2a, (4ac-b & sup2;) / 4A)"
a^x=a^1=a < a+1
There are two solutions of a ^ x = - x ^ 2 + 2x + a (a > 0, and a ≠ 0)

The number of solutions of the equation | x + 1 | + | 2x-1 | = 0 is

If the absolute value is 0, then both absolute values are 0
x+1=0
x=-1
2x-1=0
2x=1
x=0.5
If x = 0.5 and x = - 1, there is obviously no solution