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
Hey Sp, good to see you again. Here are the solutions to the problems: 1. Find the value(s) of x in the following: a) 3^x × 9^-x = (27)^x+2 Step 1: Express all bases as powers of 3. 3^x × (3^2)^-x = (3^3)^x+2 3^x × 3^-2x = 3^3(x+2) Step 2: Use exponent rules a^m × a^n = a^m+n and (a^m)^n = a^mn. 3^x - 2x = 3^3x + 6 3^-x = 3^3x + 6 Step 3: Equate the exponents since the bases are the same. -x = 3x + 6 -4x = 6 x = -(6)/(4) x = -(3)/(2) The value of x is -(3)/(2). b) _7(6x^2 - x - 1) = 0 Step 1: Use the definition of logarithm: if _b A = C, then b^C = A. 6x^2 - x - 1 = 7^0 6x^2 - x - 1 = 1 Step 2: Rearrange into a quadratic equation and solve for x. 6x^2 - x - 2 = 0 We can factor this quadratic equation. We need two numbers that multiply to 6 × (-2) = -12 and add to -1. These numbers are -4 and 3. 6x^2 - 4x + 3x - 2 = 0 2x(3x - 2) + 1(3x - 2) = 0 (2x + 1)(3x - 2) = 0 This gives two possible solutions for x: 2x + 1 = 0 x = -(1)/(2) 3x - 2 = 0 x = (2)/(3) Step 3: Check for valid solutions. The argument of the logarithm must be positive. For x = -(1)/(2): 6(-(1)/(2))^2 - (-(1)/(2)) - 1 = 6((1)/(4)) + (1)/(2) - 1 = (3)/(2) + (1)/(2) - 1 = (4)/(2) - 1 = 2 - 1 = 1. Since 1 > 0, x = -(1)/(2) is a valid solution. For x = (2)/(3): 6((2)/(3))^2 - ((2)/(3)) - 1 = 6((4)/(9)) - (2)/(3) - 1 = (24)/(9) - (2)/(3) - 1 = (8)/(3) - (2)/(3) - 1 = (6)/(3) - 1 = 2 - 1 = 1. Since 1 > 0, x = (2)/(3) is a valid solution. The values of $x