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: Rewrite the equation using a substitution. Notice that ((3)/(2))^x = (((2)/(3))^-1)^x = ((2)/(3))^-x. Let y = ((2)/(3))^x. Then the equation becomes: y + y^-1 = (13)/(6) y + (1)/(y) = (13)/(6) Step 2: Convert the equation into a quadratic form. Multiply the entire equation by 6y to eliminate the denominators: 6y · y + 6y · (1)/(y) = 6y · (13)/(6) 6y^2 + 6 = 13y Rearrange the terms to form a standard quadratic equation: 6y^2 - 13y + 6 = 0 Step 3: Solve the quadratic equation for y. We can factor the quadratic equation. We need two numbers that multiply to 6 × 6 = 36 and add up to -13. These numbers are -4 and -9. 6y^2 - 9y - 4y + 6 = 0 Factor by grouping: 3y(2y - 3) - 2(2y - 3) = 0 (3y - 2)(2y - 3) = 0 This gives two possible values for y: 3y - 2 = 0 3y = 2 y = (2)/(3) 2y - 3 = 0 2y = 3 y = (3)/(2) Step 4: Substitute back y = ((2)/(3))^x and solve for x. Case 1: y = (2)/(3) ((2)/(3))^x = (2)/(3) Since (2)/(3) = ((2)/(3))^1, we have: ((2)/(3))^x = ((2)/(3))^1 Therefore, x = 1. Case 2: y = (3)/(2) ((2)/(3))^x = (3)/(2) Since (3)/(2) is the reciprocal of (2)/(3), we can write (3)/(2) = ((2)/(3))^-1. ((2)/(3))^x = ((2)/(3))^-1 Therefore, x = -1. The values of x are 1 and -1. x = 1, -1 Send me the next one 📸