Solve the wave equation, and the binomial function.

Part 1: General Solution of the Wave Equation Step 1: The given wave equation is a one-dimensional wave equation. (^2 y)/( t^2) = c^2 (^2 y)/( x^2) The general solution to this equation is given by d'Alembert's formula, which represents the superposition of two waves traveling in opposite directions. y(x,t) = f(x - ct) + g(x + ct) where f and g are arbitrary twice-differentiable functions. The general solution is y(x,t) = f(x - ct) + g(x + ct). Part 2: Quadratic Function Analysis Given F(x) = ax^2 + bx + c and F(x-1) + F(x+1) = 3x^2 + 5x - 12. Step 1: Expand F(x-1) and F(x+1). F(x-1) = a(x-1)^2 + b(x-1) + c = a(x^2 - 2x + 1) + bx - b + c = ax^2 - 2ax + a + bx - b + c F(x+1) = a(x+1)^2 + b(x+1) + c = a(x^2 + 2x + 1) + bx + b + c = ax^2 + 2ax + a + bx + b + c Step 2: Sum F(x-1) and F(x+1) and equate coefficients. F(x-1) + F(x+1) = (ax^2 - 2ax + a + bx - b + c) + (ax^2 + 2ax + a + bx + b + c) = 2ax^2 + 2bx + 2a + 2c Comparing this to 3x^2 + 5x - 12: For the x^2 term: 2a = 3 a = (3)/(2) For the x term: 2b = 5 b = (5)/(2) For the constant term: 2a + 2c = -12 Substitute a = (3)/(2): 2((3)/(2)) + 2c = -12 3 + 2c = -12 2c = -15 c = -(15)/(2) Step 3: State F(x). Using the values a = (3)/(2), b = (5)/(2), and c = -(15)/(2): F(x) = (3)/(2)x^2 + (5)/(2)x - (15)/(2) The function is F(x) = (3)/(2)x^2 + (5)/(2)x - (15)/(2). Step 4: Calculate F(2), F(3), F(-2), and F(-3). F(2) = (3)/(2)(2)^2 + (5)/(2)(2) - (15)/(2) = (3)/(2)(4) + 5 - (15)/(2) = 6 + 5 - (15)/(2) = 11 - (15)/(2) = (22-15)/(2) = (7)/(2) F(3) = (3)/(2)(3)^2 + (5)/(2)(3) - (15)/(2) = (3)/(2)(9) + (15)/(2) - (15)/(2) = (27)/(2) F(-2) = (3)/(2)(-2)^2 + (5)/(2)(-2) - (15)/(2) = (3)/(2)(4) - 5 - (15)/(2) = 6 - 5 - (15)/(2) = 1 - (15)/(2) = (2-15)/(2) = -(13)/(2) F(-3) = (3)/(2)(-3)^2 + (5)/(2)(-3) - (15)/(2) = (3)/(2)(9) - (15)/(2) - (15)/(2) = (27)/(2) - (30)/(2) = -(3)/(2) Step 5: Determine if F(x) has a minimum or maximum and find its coordinates. Since the coefficient of x^2 is a = (3)/(2), which is positive (a > 0), the parabola opens upwards. Therefore, F(x) has a minimum value. The x-coordinate of the vertex (the minimum point) is given by x = -(b)/(2a). x = -(5/2)/(2(3/2)) = -(5/2)/(3) = -(5)/(6) Now, substitute this x-value back into F(x) to find the y-coordinate (the minimum value): F(-(5)/(6)) = (3)/(2)(-(5)/(6))^2 + (5)/(2)(-(5)/(6)) - (15)/(2) = (3)/(2)((25)/(36)) - (25)/(12) - (15)/(2) = (25)/(24) - (50)/(24) - (180)/(24) = (25 - 50 - 180)/(24) = (-205)/(24) The function F(x) has a minimum value. The coordinates of the minimum point are (-(5)/(6), -(205)/(24)). Send me the next one 📸