вправа 2.56 гдз 11 клас алгебра Істер 2019
Вправа 2.56
Умова:
Розв'яжіть рівняння:
1) (5 + 2√6)х + (5 - 2√6)х = 10;
2) 9(3 - √8)4х+1 = 3/(3 + √8)4х+1;
3) (√2 - 1)х + (√2 + 1)х = 6.
1) (5 + 2√6)х + (5 - 2√6)х = 10;
2) 9(3 - √8)4х+1 = 3/(3 + √8)4х+1;
3) (√2 - 1)х + (√2 + 1)х = 6.
Відповідь:
1) (5 + 2√6)х + (5 - 2√6)х = 10
(5 + 2√6) • (5 - 2√6) = 25 - 24 = 1, =>
(5 - 2√6) = 1/(5 + 2√6)
нехай: (5 - 2√6)х = t, t > 0
t + 1/t - 10 = 0
t2 - 10t + 1 = 0
Д = (-10)2 - 4 = 100 - 4 = 96
√96 = √16•6 = 4√6
t1;2 = (10±4√6)/2 = (2(5±2√6))/2
t1 = 5 + 2√6 t2 = 5 - 2√6
(5 - 2√6)х = 5 + 2√6 (5 - 2√6)х = 5 - 2√6
х = -1 х = 1;
(5 + 2√6) • (5 - 2√6) = 25 - 24 = 1, =>
(5 - 2√6) = 1/(5 + 2√6)
нехай: (5 - 2√6)х = t, t > 0
t + 1/t - 10 = 0
t2 - 10t + 1 = 0
Д = (-10)2 - 4 = 100 - 4 = 96
√96 = √16•6 = 4√6
t1;2 = (10±4√6)/2 = (2(5±2√6))/2
t1 = 5 + 2√6 t2 = 5 - 2√6
(5 - 2√6)х = 5 + 2√6 (5 - 2√6)х = 5 - 2√6
х = -1 х = 1;
2) 9(3 - √8)4х+1 = 3/(3 + √8)4х+1
(34х+1)/((3+√8)4х+1 • 9 • (3-√8)4х+1) = 1
(34х+1)/(9 • (3+√8) • (3-√8))4х+1 = 1
(34х+1)/(9 • (9 - 8)4х+1) = 1
(34х+1)/(9 • 14х+1) = 1
34х+1 = 9
34х+1 = 32
4х + 1 = 2
4х = 2 - 1
4х = 1
х = 0,25;
(34х+1)/((3+√8)4х+1 • 9 • (3-√8)4х+1) = 1
(34х+1)/(9 • (3+√8) • (3-√8))4х+1 = 1
(34х+1)/(9 • (9 - 8)4х+1) = 1
(34х+1)/(9 • 14х+1) = 1
34х+1 = 9
34х+1 = 32
4х + 1 = 2
4х = 2 - 1
4х = 1
х = 0,25;
3) (√2 - 1)х + (√2 + 1)х = 6
(√2 - 1)(√2 + 1) = 2 - 1 = 1 =>
√2 - 1 = 1/(√2 + 1)
нехай: (√2 - 1)х = t, t > 0
t + 1/t - 6 = 0
t2 - 6t + 1 = 0
Д = (-6)2 - 4 • 1 = 36 - 4 = 32
√Д = √32 = √16•2 = 4√2
t1;2 = (6±4√2)/2 = (2(3±2√2))/2 = 3±2√2
(√2 - 1)х = t1 (√2 - 1)х = t2
(√2 - 1)х = 3 + 2√2 (√2 - 1)х = 3 - 2√2
(√2 - 1)х = (√2)2 + 2√2 + 1 (√2 - 1)х = (√2)2 - 2√2 + 1
(√2 - 1)х = (√2 + 1)2 (√2 - 1)х = (√2 - 1)2
х = -2 х = 2.