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: Let y = _3 x. Substitute this into the given equation. The equation is (_3 x)^2 - (1)/(2) _3 x = (3)/(2). Substituting y = _3 x, we get: y^2 - (1)/(2)y = (3)/(2) Step 2: Rearrange the equation into a standard quadratic form ay^2 + by + c = 0. Multiply the entire equation by 2 to eliminate fractions: 2y^2 - y = 3 Subtract 3 from both sides: 2y^2 - y - 3 = 0 Step 3: Solve the quadratic equation for y. We can factor the quadratic equation. We need two numbers that multiply to 2 × (-3) = -6 and add up to -1. These numbers are -3 and 2. 2y^2 - 3y + 2y - 3 = 0 Factor by grouping: y(2y - 3) + 1(2y - 3) = 0 (y + 1)(2y - 3) = 0 This gives two possible values for y: y + 1 = 0 y = -1 2y - 3 = 0 2y = 3 y = (3)/(2) Step 4: Substitute back _3 x for y and solve for x. Case 1: y = -1 _3 x = -1 By the definition of a logarithm (b^y = x _b x = y), we have: x = 3^-1 x = (1)/(3) Case 2: y = (3)/(2) _3 x = (3)/(2) By the definition of a logarithm, we have: x = 3^3/2 x = sqrt(3^3) x = sqrt(27) x = sqrt(9 × 3) x = 3sqrt(3) Both solutions are valid since x > 0 for _3 x. The solutions for x are (1)/(3) and 3sqrt(3). x = (1)/(3), 3sqrt(3) Send me the next one 📸