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Quadratic Equation

Problem: Solve the quadratic equation: 2x² - 5x + 3 = 0

Solution:

                        2x² - 5x + 3 = 0

                        Using the quadratic formula:
                        x = (-b ± √(b² - 4ac)) / 2a
                        where a = 2, b = -5, c = 3

                        Discriminant: Δ = (-5)² - 4 × 2 × 3 = 25 - 24 = 1

                        x = [5 ± √1] / 4
                        x = [5 ± 1] / 4

                        Solutions:
                        x₁ = (5 + 1) / 4 = 6 / 4 = 1.5
                        x₂ = (5 - 1) / 4 = 4 / 4 = 1
                        

Integral Calculus

Problem: Calculate the indefinite integral ∫(3x² - 5x + 2)dx

Solution:

                        ∫(3x² - 5x + 2)dx = x³ - (5/2)x² + 2x + C
                        

Matrix Multiplication

Problem: Find the product of the matrices:
A = [2, 3] and B = [1, 2]
[4, 5] [3, 4]

Solution:

                        Matrix multiplication formula: C = A × B

                        C = [2×1 + 3×3, 2×2 + 3×4]  
                            [4×1 + 5×3, 4×2 + 5×4]

                        C = [11, 16]  
                            [19, 26]
                        

Limit Calculation

Problem: Calculate the limit: lim(x → 0) (sin(x) / x)

Solution:

                        lim(x → 0) (sin(x) / x) = 1
                        

Differential Equation

Problem: Solve the differential equation: dy/dx = 3x² - 2x + 1

Solution:

                        dy/dx = 3x² - 2x + 1

                        Integrating both sides:
                        y = x³ - x² + x + C
                        

Permutations and Combinations

Problem: From a set of 10 different balls, choose 3 balls. Find the number of ways to choose.

Solution:

                        This is a combination problem: C(10, 3) = 10! / (3! × (10 - 3)!)

                        C(10, 3) = 10 × 9 × 8 / (3 × 2 × 1) = 120
                        

Complex Numbers

Problem: Find the product of the complex numbers (3 + 4i) and (1 - 2i).

Solution:

                        (3 + 4i)(1 - 2i) = 3 × 1 + 3 × (-2i) + 4i × 1 + 4i × (-2i)

                        = 3 - 6i + 4i - 8i²

                        Since i² = -1, we get:

                        = 3 - 6i + 4i + 8

                        = 11 - 2i
                        

Triple Integral

Problem: Calculate the triple integral: ∫∫∫_V (x + y + z) dV, where V is the unit cube [0, 1] × [0, 1] × [0, 1]

Solution:

                        ∫∫∫_V (x + y + z) dV = ∫∫∫_0^1 (x + y + z) dx dy dz

                        By integrating, the result is 3/2.
                        

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