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
Here's the solution for number 7: Step 1: Rewrite the exponential equation using the property (a^m)^n = a^mn. The given equation is 5^2x - 6 × 5^x + 5 = 0. We can rewrite 5^2x as (5^x)^2. So the equation becomes: (5^x)^2 - 6(5^x) + 5 = 0 Step 2: Introduce a substitution to transform the equation into a quadratic form. Let y = 5^x. Substitute y into the equation: y^2 - 6y + 5 = 0 Step 3: Solve the quadratic equation for y. This is a quadratic equation that can be factored. We need two numbers that multiply to 5 and add up to -6. These numbers are -1 and -5. (y - 1)(y - 5) = 0 This gives two possible values for y: y - 1 = 0 y = 1 y - 5 = 0 y = 5 Step 4: Substitute back y = 5^x and solve for x. Case 1: y = 1 5^x = 1 Since any non-zero number raised to the power of 0 is 1, we can write 1 as 5^0. 5^x = 5^0 Therefore, x = 0 Case 2: y = 5 5^x = 5 Since 5 can be written as 5^1. 5^x = 5^1 Therefore, x = 1 The values of x that satisfy the equation are x = 0 or x = 1. Last free one today — make it count tomorrow, or type /upgrade for unlimited.