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: 5^x+1 = 26 - 5^1-x Step 1: Apply exponent rules to rewrite the terms. We use the rules a^m+n = a^m × a^n and a^m-n = a^m × a^-n = (a^m)/(a^n). So, 5^x+1 can be written as 5^x × 5^1 = 5 × 5^x. And 5^1-x can be written as 5^1 × 5^-x = (5)/(5^x). Substituting these into the original equation: 5 × 5^x = 26 - (5)/(5^x) Step 2: Introduce a substitution. Let y = 5^x. Since 5^x is always positive, y > 0. The equation becomes: 5y = 26 - (5)/(y) Step 3: Solve the equation for y. Multiply the entire equation by y to eliminate the fraction: 5y × y = 26 × y - (5)/(y) × y 5y^2 = 26y - 5 Rearrange the terms to form a standard quadratic equation: 5y^2 - 26y + 5 = 0 Step 4: Factor the quadratic equation. We need two numbers that multiply to 5 × 5 = 25 and add up to -26. These numbers are -1 and -25. Rewrite the middle term: 5y^2 - 25y - y + 5 = 0 Factor by grouping: 5y(y - 5) - 1(y - 5) = 0 (5y - 1)(y - 5) = 0 This gives two possible values for y: 5y - 1 = 0 5y = 1 y = (1)/(5) y - 5 = 0 y = 5 Step 5: Substitute back y = 5^x and solve for x. Case 1: y = (1)/(5) 5^x = (1)/(5) Since (1)/(5) = 5^-1, we have: 5^x = 5^-1 Therefore, x = -1 Case 2: y = 5 5^x = 5 Since 5 = 5^1, we have: 5^x = 5^1 Therefore, x = 1 The solutions for x are -1 and 1. The final answers are x=-1, x=1. What's next?