16^ (x+1/4) - 41*4^ (x-1) + 9=0

16 ^{(x + 1/4)} - 41 * 4 ^{(x - 1)} + 9 = 0;

(2⁴) ^{(x + 1/4)} - (40 + 1) * (2²) ^{(x - 1)} + 9 = 0;

2^{4 * (x + 1/4)} - 40 * 2^{2 * (x - 1)} - 2^{2 * (x - 1)} + 9 = 0;

2^{4x + 1} - 2² * 10 * 2^{2*x - 2} - 2^{2x - 2} + 9 = 0;

2 * 2^4x - 10 * 2^{2x - 2+2} - 0,25 * 2^2x + 9 = 0;

2 * 2^4x - 10 * 2^2x - 0,25 * 2^2x + 9 = 0;

2 * (2^{2 х}) ² - 10,25 * 2^2x + 9 = 0;

Пусть 2^2x = р;

2 * р² - 10,25 * р + 9 = 0;

Уравнение вида: a * x² + b * x + c = 0, где а = 2; b = - 10,25; с = 9; х = р, может иметь 2 решения:

р₁ = ( - b - √‾ (b² - 4 * a * c)) / (2 * a) = (10,25 - √‾ ((-10,25) ² - 4 * 2 * 9)) / (2 * 2) = (10,25 - √‾ (105,0625 - 72)) / 4 = (10,25 - √‾ (33,0625)) / 4 = (10,25 - 5,75) / 4 = 4,5/4 = 1,125.

р₂ = ( - b + √‾ (b² - 4 * a * c)) / (2 * a) = (10,25 + √‾ ((-10,25) ² - 4 * 2 * 9)) / (2 * 2) = (10,25 + √‾ (105,0625 - 72)) / 4 = (10,25 + √‾ (33,0625)) / 4 = (10,25 + 5,75) / 4 = 16/4 = 4.

р₁ = (2 ^x) ² = 1,125;

2^2x = 4^x = 1,125 = 9/8;

х = log_9/8 4 = log_1.125 4;

р₂ = (2^x) ² = 4 = 2²;

2^2x = 2²;

2 * х = 2;

х = 2/2 = 1.