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 terms in the equation using exponent rules. Recall that a^mn = (a^m)^n and a^m+n = a^m · a^n. 2^2x - 3(2^x+2) + 32 = 0 (2^x)^2 - 3(2^x · 2^2) + 32 = 0 (2^x)^2 - 3(4 · 2^x) + 32 = 0 (2^x)^2 - 12(2^x) + 32 = 0 Step 2: Let y = 2^x to transform the equation into a quadratic form. y^2 - 12y + 32 = 0 Step 3: Solve the quadratic equation for y. We can factor the quadratic expression. We need two numbers that multiply to 32 and add up to -12. These numbers are -4 and -8. (y - 4)(y - 8) = 0 This gives two possible values for y: y - 4 = 0 y = 4 y - 8 = 0 y = 8 Step 4: Substitute back y = 2^x and solve for x. Case 1: y = 4 2^x = 4 Since 4 = 2^2, we have: 2^x = 2^2 x = 2 Case 2: y = 8 2^x = 8 Since 8 = 2^3, we have: 2^x = 2^3 x = 3 The solutions for x are 2 and 3. x=2, 3 That's 2 down. 3 left today — send the next one.