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
Fresh day Mina, let's solve. The equation from the image is: 9^x - 10 × 3^x + 9 = 0 Step 1: Rewrite the terms with a common base. We know that 9 = 3^2. So, 9^x can be written as (3^2)^x = 3^2x = (3^x)^2. Substituting this into the equation: (3^x)^2 - 10 × 3^x + 9 = 0 Step 2: Introduce a substitution to simplify the equation. Let y = 3^x. Substituting y into the equation transforms it into a quadratic equation: y^2 - 10y + 9 = 0 Step 3: Solve the quadratic equation for y. We can factor this quadratic equation. We need two numbers that multiply to 9 and add up to -10. These numbers are -1 and -9. (y - 1)(y - 9) = 0 This gives two possible values for y: y - 1 = 0 y = 1 y - 9 = 0 y = 9 Step 4: Substitute back y = 3^x and solve for x. Case 1: y = 1 3^x = 1 Since any non-zero number raised to the power of 0 is 1, we have 3^0 = 1. Therefore, x = 0 Case 2: y = 9 3^x = 9 Since 9 = 3^2, we can write: 3^x = 3^2 Therefore, x = 2 The solutions for x are 0 and 2. The final answers are x=0, x=2. Got more? Send 'em!