This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Simplify the given equation by evaluating the powers of i. Recall that i^2 = -1 and i^4 = 1. The given equation is: 5x^2i - 4 = 3yi^4 + y^2i^2 - y(xi + i) Substitute the values of i^4 and i^2: 5x^2i - 4 = 3y(1) + y^2(-1) - yxi - yi 5x^2i - 4 = 3y - y^2 - yxi - yi Step 2: Rearrange the equation to group real and imaginary parts. Left side: -4 + 5x^2i Right side: (3y - y^2) + (-yx - y)i Equate the real parts: -4 = 3y - y^2 Equate the imaginary parts: 5x^2 = -yx - y Step 3: Solve the equation from the real parts for y. -4 = 3y - y^2 y^2 - 3y - 4 = 0 Factor the quadratic equation: (y - 4)(y + 1) = 0 This gives two possible values for y: y = 4 or y = -1. The problem states that y N, which means y must be a natural number (a positive integer). Therefore, we choose the positive value for y. y = 4 Step 4: Substitute the value of y into the equation from the imaginary parts and solve for x. 5x^2 = -yx - y Substitute y = 4: 5x^2 = -4x - 4 Rearrange into a standard quadratic form: 5x^2 + 4x + 4 = 0 To solve for x, use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=5, b=4, c=4. x = -4 ± sqrt(4^2 - 4(5)(4))2(5) x = -4 ± sqrt(16 - 80)10 x = -4 ± sqrt(-64)10 Since sqrt(-64) = sqrt(64 × -1) = 8i: x = (-4 ± 8i)/(10) Simplify the expression: x = (-2 ± 4i)/(5) The solutions are: y = 4 x = -(2)/(5) + (4)/(5)i or -(2)/(5) - (4)/(5)i Drop the next question.