Y = the square of X + 2ax-1, find the range of a when x belongs to - 1 to 3 closed interval

Y = the square of X + 2ax-1, find the range of a when x belongs to - 1 to 3 closed interval

y=x^2+2ax-1=(x+a)^2-a^2-1
When - a = 1, the minimum value f (- 1) = - 2A
Maximum f (3) = 6A + 8
When - 1

Y = x ^ 2-2ax + 1, X is in the range of [- 1,1], find the range

[interpretation]
y＝x²－2ax＋1＝(x-a)²＋(1-a²)
X ∈ [- 1,1], where (x-a) ∈ [- 1-A, 1-A]
Discussion:
(1) A ＜ - 1, then 0 ＜ - 1-A ＜ 1-A, there is
When x-a = - 1-A, y (min) = (1 + a) & sup2; + (1-A & sup2;) = 2 + 2A
When x-a = 1-A, y (max) = (1-A) & sup2; + (1-A & sup2;) = 2-2a
(2) If - 1 ≤ a < 0, then - 1-A ≤ 0 < 1-A, and | - 1-A | = 1 + a < 1-A = | - 1-A |, there is
When x-a = 0, y (min) = 1-A & sup2;
When x-a = 1-A, y (max) = (1-A) & sup2; + (1-A & sup2;) = 2-2a
(2) 0 ≤ a < 1, then - 1-A < 0 < 1-A, and | - 1-A | = 1 + a ≥ 1-A = | - 1-A |
When x-a = 0, y (min) = 1-A & sup2;
When x-a = - 1-A, y (max) = 2 + 2A
(4) A ≥ 1, then - 1-A < 1-A ≤ 0, there is
When x-a = 1-A, y (min) = (1-A) & sup2; + (1-A & sup2;) = 2-2a
When x-a = - 1-A, y (max) = (1 + a) & sup2; + (1-A & sup2;) = 2 + 2A
To sum up, there are:
When a ∈ (- ∞, - 1), the range is [2 + 2a, 2-2a];
When a ∈ [- 1,0), the range [1-A & sup2;, 2-2a];
When a ∈ [0,1], the range [1-A & sup2;, 2 + 2A];
When a ∈ [1, + ∞), the range is [2-2a, 2 + 2A]

Function f (x) = x ^ 2 + 2aX + 3, X ∈ (- 4,4). When a = - 1, find the range of function f (x). Function has zero on (- 4,4), find the range of A

(1) when a = - 1:
f(x)=x^2-2x+3=(x-1)^2+2≥2
The opening is upward, and the axis of symmetry x = 1 is between the interval (- 4,4)
When x = 1, there is a minimum of 2
f(-4)=5^2+2=27
f(4)=3^2+2=11
The range of F (x) on the interval (- 4,4) [2,27]
⑵
F (x) = x ^ 2 + 2aX + 3 has zero,
Discriminant = (2a) ^ 2-4 * 3 = 4A ^ 2-12 ≥ 0
A ≤ - radical 3, or ≥ radical 3. (1)
X = [- 2A ± 2 radical (4a ^ 2-12)] / 2 = - a ± radical (a ^ 2-3)
F (x) = x ^ 2 + 2aX + 3 has zero on (- 4,4)
-4 ≤ = - a-radical (a ^ 2-3) ≤ 4. (2)
perhaps
-4 ≤ = - A + radical (a ^ 2-3) ≤ 4. (3)
From (2), it is concluded that:
Radical (a ^ 2-3) ≤ 4-A, - 3 ≤ 16-8a, a ≤ 19 / 8
Radical (a ^ 2-3) ≥ (4 + a), - 3 ≥ 16 + 8a, a ≤ - 19 / 8
∴a≤-19/8
From (3), it is concluded that:
Radical (a ^ 2-3) ≥ A-4, - 3 ≥ - 8A + 16, a ≥ 19 / 8
Radical (a ^ 2-3) ≤ 4 + A, - 3 ≤ 16 + 8a, a ≥ - 19 / 8
∴a≥19/8
Both a ≤ - 19 / 8 and a ≥ 19 / 8 meet the requirements of (1)
A ≤ - 19 / 8 and a ≥ 19 / 8

If two univariate quadratic equations x2 + X + a = 0 and X2 + ax + 1 = 0 with real coefficients about X have a common real root, then a=______ ．

If A-1 = 0, i.e. a = 1, the b2-4ac of the equations x2 + X + a = 0 and X2 + ax + 1 = 0 are less than 0, i.e. the equation has no solution; therefore, a ≠ 1, and the common root is x = 1. Substituting x = 1 into the equation has: 1 + 1 + a = 0, a = - 2