4.1. (log2 (x+1) - 3)*√(x - a) = 0
У этого уравнения два корня:
1) log2 (x+1) = 3
x + 1 = 2^3 = 8
x = 7
2) x = a.
Но при а = 7 эти корни совпадают и получается один корень.
ответ: при а = 7 - один корень x = 7.
При а ≠ 7 - два корня, x1 = 7; x2 = a.
4.2. (2sin x + 1)(2cos y + 3) ≥ 15
Отметим, что sin x € [-1; 1]; 2sin x + 1 € [-2+1; 2+1] = [-1; 3]
cos y € [-1; 1]; 2cos y + 3 € [-2+3; 2+3] = [1; 5].
Чтобы произведение
(2sin x + 1)(2cos y + 3) = 15,
должно быть
{ 2sin x + 1 = 3
{ 2cos y + 3 = 5
То есть должно быть
{ sin x = 1
{ cos y = 1
x = Π/2 + 2Πk, k € Z
y = 2Πn, n € Z.
Вот такие точки и надо отметить на плоскости.
Simplifying
4x3 + -81x = 0
Reorder the terms:
-81x + 4x3 = 0
Solving
-81x + 4x3 = 0
Solving for variable 'x'.
Factor out the Greatest Common Factor (GCF), 'x'.
x(-81 + 4x2) = 0
Factor a difference between two squares.
x((9 + 2x)(-9 + 2x)) = 0
Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:
Simplifying
x = 0
Solving
x = 0
Move all terms containing x to the left, all other terms to the right.
Simplifying
x = 0
Subproblem 2
Set the factor '(9 + 2x)' equal to zero and attempt to solve:
Simplifying
9 + 2x = 0
Solving
9 + 2x = 0
Move all terms containing x to the left, all other terms to the right.
Add '-9' to each side of the equation.
9 + -9 + 2x = 0 + -9
Combine like terms: 9 + -9 = 0
0 + 2x = 0 + -9
2x = 0 + -9
Combine like terms: 0 + -9 = -9
2x = -9
Divide each side by '2'.
x = -4.5
Simplifying
x = -4.5
Subproblem 3
Set the factor '(-9 + 2x)' equal to zero and attempt to solve:
Simplifying
-9 + 2x = 0
Solving
-9 + 2x = 0
Move all terms containing x to the left, all other terms to the right.
Add '9' to each side of the equation.
-9 + 9 + 2x = 0 + 9
Combine like terms: -9 + 9 = 0
0 + 2x = 0 + 9
2x = 0 + 9
Combine like terms: 0 + 9 = 9
2x = 9
Divide each side by '2'.
x = 4.5
Simplifying
x = 4.5
Solution
x = {0, -4.5, 4.5}
