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: Rewrite the equation using a substitution. The given equation is 2(5^2x) - 7(5^x) = 15. Let y = 5^x. Then 5^2x = (5^x)^2 = y^2. Substitute y into the equation: 2y^2 - 7y = 15 Step 2: Rearrange the equation into standard quadratic form. Subtract 15 from both sides: 2y^2 - 7y - 15 = 0 Step 3: Solve the quadratic equation for y. We can factor the quadratic equation. We need two numbers that multiply to 2 × (-15) = -30 and add up to -7. These numbers are -10 and 3. Rewrite the middle term: 2y^2 - 10y + 3y - 15 = 0 Factor by grouping: 2y(y - 5) + 3(y - 5) = 0 (2y + 3)(y - 5) = 0 This gives two possible solutions for y: 2y + 3 = 0 2y = -3 y = -(3)/(2) y - 5 = 0 y = 5 Step 4: Substitute back y = 5^x and solve for x. Recall that y = 5^x. Since 5^x must always be a positive value for real x, we discard the negative solution for y. Therefore, y = 5 is the only valid solution. Set 5^x equal to 5: 5^x = 5 Since 5 = 5^1, we can write: 5^x = 5^1 Equating the exponents, we find: x = 1 Step 5: Compare the solution with the given options. The calculated value x=1 matches option D. The final answer is D. 1.