Решите уравнение:
а) (х + 4)2 = 3х + 40; б) (2х - 3)2 = 11х - 19;
в) 3(х + 4)2 = 10x + 82; г) 15х2 + 17 = 15(х + 1)2;
д) (х + 1)2 = 7918 - 2х; е) (х + 2)2 = 3131 - 2х;
ж) (x + 1)2 = (2х - 1)2; з) (х - 2)2 + 48 = (2 - 3х)2.
Решение:
а) (х + 4)2 = 3х + 40; х2 + 8х + 16 - 3х-40 = 0; х2 + 5х - 24 = 0;
D = 25 + 4 • 24 = 25 + 96 = 121; х = (-5±11)/2; x1 = -8; х2 = 3;
б) (2х - 3)2 = 11х - 19; 4х2 - 12х + 9 - 11х + 19 = 0; 4х2 - 23х + 28 = 0;
D = 232 - 4 • 4 • 28 = 529 - 448 = 81; х = (23±9)/8; x1 = 1,75; х2 = 4;
в) 3(х + 4)2 = 10x + 82; 3х2 + 24х + 48 = 10х + 32; 3х2 + 14х + 16 = 0;
D = 72 - 3 • 16 = 49 - 48 = 1; х = (-7±1)/3; x1 = - 2 2/3; х2 = - 2;
г) 15х2 + 17 = 15(х + 1)2; 15х2+ 17 = 15х2 + 30х + 15; 30x = 2; x = 1/15;
д) (х + 1)2 = 7918 - 2х; х2 + 2х + 1 - 7918 + 2х = 0; х2 + 4х - 7917 = 0;
D1 = 22 + 7917 = 7921; х = -2 ± 89; x1 = -91; х2 = 87;
е) (х + 2)2 = 3131 - 2х; х2 + 4х + 4 - 3131 + 2х = 0; х2 + 6х - 3127 = 0;
D1 = 32 + 3127 = 3136; х = -3 ± 56; x1 = -59; х2 = 53;
ж) (x + 1)2 = (2х - 1)2; (2х - 1)2 - (х + 1)2 = 0; (2х - 1 - х - 1) (2х - 1 + x + 1) = 0; (х - 2) • х = 0; х2 - 2х = 0; х(х - 2) = 0; x1 = 0; х2 = 2;
з) (х - 2)2 + 48 = (2 - 3х)2; х2 - 4х + 4 + 48 = 4 - 12х + 9х2; 8х2 - 8х - 48 = 0; х2 - х - б = 0;
D = 1 + 24 = 25; х = (1±5)/2; x1 = -2; х2 = 3.
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