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
Right sampomah776, let's go. Here's the solution for question 4. Step 1: Calculate the product AB. Given A = x^2 & 3 \\ 1 & 3x and B = 3 & 6 \\ 2 & x . AB = x^2 & 3 \\ 1 & 3x 3 & 6 \\ 2 & x AB = (x^2)(3) + (3)(2) & (x^2)(6) + (3)(x) \\ (1)(3) + (3x)(2) & (1)(6) + (3x)(x) AB = 3x^2 + 6 & 6x^2 + 3x \\ 3 + 6x & 6 + 3x^2 Step 2: Calculate the product BA. BA = 3 & 6 \\ 2 & x x^2 & 3 \\ 1 & 3x BA = (3)(x^2) + (6)(1) & (3)(3) + (6)(3x) \\ (2)(x^2) + (x)(1) & (2)(3) + (x)(3x) BA = 3x^2 + 6 & 9 + 18x \\ 2x^2 + x & 6 + 3x^2 Step 3: Set AB = BA and equate the corresponding elements. 3x^2 + 6 & 6x^2 + 3x \\ 3 + 6x & 6 + 3x^2 = 3x^2 + 6 & 9 + 18x \\ 2x^2 + x & 6 + 3x^2 Equating the top-left and bottom-right elements gives identities (3x^2+6 = 3x^2+6 and 6+3x^2 = 6+3x^2), which do not help find x. We need to equate the other two elements: From the top-right elements: 6x^2 + 3x = 9 + 18x From the bottom-left elements: 3 + 6x = 2x^2 + x Step 4: Solve the quadratic equation(s). Let's solve the equation from the top-right elements: 6x^2 + 3x - 18x - 9 = 0 6x^2 - 15x - 9 = 0 Divide the entire equation by 3: 2x^2 - 5x - 3 = 0 We can solve this quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -(-5) ± sqrt((-5)^2 - 4(2)(-3))2(2) x = 5 ± sqrt(25 + 24)4 x = 5 ± sqrt(49)4 x = (5 ± 7)/(4) This gives two possible values for x: x_1 = (5 + 7)/(4) = (12)/(4) = 3 x_2 = (5 - 7)/(4) = (-2)/(4) = -(1)/(2) Now, let's check the equation from the bottom-left elements: 3 + 6x = 2x^2 + x 2x^2 + x - 6x - 3 = 0 2x^2 - 5x - 3 = 0 This is the same quadratic equation, so it yields the same solutions. The possible values of x are 3, -(1)/(2). What's next?