Solve 4ˣ + 64x - 192 = 0 So, 4ˣ + 4³x - 3*4³ = 0 So, 4⁽ˣ⁻ ³⁾ + x - 3 = 0 So, 4⁽ˣ⁻ ³⁾ = 3 - x So, 2²⁽ˣ⁻ ³⁾ = 3 - x = -2(x - 3)/2 Let t = 2(x - 3) So, 2ᵗ = - t/2 By observation t = -1 is a solution. So x = 2.5
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Math Olympiad: 4ˣ + 64x - 192 = 0; x = ? 4ˣ = 192 - 64x = 4³(3 - x), 4ˣ/4³ = 4ˣ⁻³ = 3 - x Trial-and-error solution; 4ˣ⁻³ + x = 3 > 0; 3 > x > 1 x = 2: 4ˣ⁻³ + x = 4²⁻³ + 2 = 1/4 + 2 < 3; x > 2 x = 3: 4³⁻³ + 3 = 4⁰ + 3 = 4 > 3; 3 > x > 2; In-between x = 2 + 1/2 = 5/2: 4⁽⁵⸍²⁾⁻³ + 5/2 = 4⁻¹⸍² + 5/2 = 1/2 + 5/2 = 3; Proved Answer check: 4ˣ + 64x - 192 = 4⁽⁵⸍²⁾ + 64(5/2) - 192 = 2⁵ + 32(5) - 192 = 0; Confirmed Final answer: x = 5/2
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