a) x² + 3x - 130 = 0
D = 3² - 4 * 1 * (-130) = 9 + 520 = 529
x₁ = -3 + 23/2 = 20/2 = 10
x₂ = -3 - 23/2 = -26/2 = -13
б) x² - 2x + 10 = 0
D = (-2)² - 4 * 1 * 10 = 4 - 40 = -36
Дискриминант равен отрицательному числу. Уравнение не имеет решений.
в) x²- 6x + 34 = 0
D = (-6)² - 4 * 1 * 34 = 36 - 136 = -100
г) 4x² - 4x = 3
4x² - 4x - 3 = 0
D = (-4)² - 4 * 4 * (-3) = 16 + 48 = 64
x₁ = 4 - 8/2 = -4/2 = -2
x₂ = 4 + 8/2 = 12/2 = 6
д) x²/2 - x/3 + 7*3/8 = 8
x²/2 - x/3 + 59/8 = 8 /*24
12x² - 8x + 177 = 192
12x² - 8x + 177 - 192 = 0
12x² - 8x - 15 = 0
D = (-8)² - 4 * 12 * (-15) = 64 + 720 = 784
x₁ = 8 - 24/24 = 16/24 = 2/3
x₂ = 8 + 24/24 = 32/24 = 4/3 = 1*1/3
е) x²/3 - 7x/18 + 4 = 5*1/9
x²/3 - 7x/18 + 4 = 46/9 / *18
6x² - 7x + 72 = 92
6x² - 7x + 72 - 92 = 0
6x² - 7x - 20 = 0
D = (-7)² - 4 * 6 * (-20) = 49 + 480 = 529
x₁ = 7 - 23/12 = -16/12 = -4/3 = -1*1/3
x₂ = 7 + 23/12 = 30/12 = 5/2 = 2*1/2
a) x² + 3x - 130 = 0
D = 3² - 4 * 1 * (-130) = 9 + 520 = 529
x₁ = -3 + 23/2 = 20/2 = 10
x₂ = -3 - 23/2 = -26/2 = -13
б) x² - 2x + 10 = 0
D = (-2)² - 4 * 1 * 10 = 4 - 40 = -36
Дискриминант равен отрицательному числу. Уравнение не имеет решений.
в) x²- 6x + 34 = 0
D = (-6)² - 4 * 1 * 34 = 36 - 136 = -100
Дискриминант равен отрицательному числу. Уравнение не имеет решений.
г) 4x² - 4x = 3
4x² - 4x - 3 = 0
D = (-4)² - 4 * 4 * (-3) = 16 + 48 = 64
x₁ = 4 - 8/2 = -4/2 = -2
x₂ = 4 + 8/2 = 12/2 = 6
д) x²/2 - x/3 + 7*3/8 = 8
x²/2 - x/3 + 59/8 = 8 /*24
12x² - 8x + 177 = 192
12x² - 8x + 177 - 192 = 0
12x² - 8x - 15 = 0
D = (-8)² - 4 * 12 * (-15) = 64 + 720 = 784
x₁ = 8 - 24/24 = 16/24 = 2/3
x₂ = 8 + 24/24 = 32/24 = 4/3 = 1*1/3
е) x²/3 - 7x/18 + 4 = 5*1/9
x²/3 - 7x/18 + 4 = 46/9 / *18
6x² - 7x + 72 = 92
6x² - 7x + 72 - 92 = 0
6x² - 7x - 20 = 0
D = (-7)² - 4 * 6 * (-20) = 49 + 480 = 529
x₁ = 7 - 23/12 = -16/12 = -4/3 = -1*1/3
x₂ = 7 + 23/12 = 30/12 = 5/2 = 2*1/2
an = a1 + d*(n - 1)
S(n) = (a1 + an)*n/2 = (2a1 + d*(n-1))*n/2
1) a1 = -5, n = 23, S(n) = 1909
1909 = (-2*5 + d*22)*23/2 = (-5 + 11d)*23
-5 + 11d = 1909/23 = 83
11d = 88, d= 8
2) a1 = -3,87, d= -2,77 + 3,87 = 1,1, n = 10
a10 = a1 + 9d = -3,87 + 9*1,1 = 9,9 - 3,87 = 6,03
S(10) = (-3,87 + 6,03)*10/2 = 2,16*5 = 10,8
3) a2 = a1 + d= 2, a9 = a1 + 8d = 6,9
a9 - a2 = 7d = 6,9 - 2 = 4,9
d= 0,7
4) 1) x1 = 3 + 2 = 5, x2 = 6 + 2 = 8, d= 3
S(20) = (2*5 + 3*19)*20/2 = (10 + 57)*10 = 670
2) x1 = 4 - 9 = -5, x2 = 8 - 9 = -1, d= 4
S(30) = (-2*5 + 4*29)*30/2 = (-10 + 116)*15 = 1590
5) 1) d= 2, an = 49, S(n) = 702
Система
{ an = a1 + d(n-1) = a1 + 2(n-1) = 49
{ S(n) = (a1 + an)*n/2 = (a1 + 49)*n/2 = 702
{ a1 + 2n = 49 + 2 = 51
{ a1*n + 49n = 702*2 = 1404
{ a1 = 51 - 2n
{ (51 - 2n)*n + 49n - 1404 = 0
-2n^2 + 100n - 1404 = 0
n^2 - 50n + 702 = 0
(n - 27)(n - 13) = 0
n = 13, a1 = 51 - 26 = 25
n = 27, a1 = 51 - 54 = -3
2) an = 18 - 2n, S(n) = n*(17 - n)
an = a1 + d(n-1) = a1-d + dn = 18 - 2n
S(n) = (2a1 + d(n-1))*n/2 = n*(17 - n)
Система
{ (a1-d) + dn = 18 - 2n
{ (2a1-d) + dn = 2(17 - n) = 34 - 2n
Из 2 уравнения вычитаем 1 уравнение
a1 = 34 - 18 = 16
Подставляем обратно в 1 уравнение
16 + dn - d = 18 - 2n
dn - d = 2 - 2n
d(n - 1) = -2(n - 1)
d= -2
Количество членов n узнать не удалось, к сожалению.