1. ) 176x = 121 + 64x ^ 2
176x - 64x ^ 2 = 121
176x - 64x ^ 2 - 121 = 0
ax ^ 2 + bx + c
a×c = - 64 × - 121 = 7744
b = 176 - 64x ^ 2 + 88x + 88x - 121
( - 64x ^ 2 + 88x) + (88x - 121)
8x ( - 8x + 11) - 11( - 8x + 11)
( - 8x + 11)(8x - 11) = 0
8x - 11 = 0 - 8x = 11
x = 11÷ 8
Forma decimal = 1, 375
Formula numérica mixta : x = 1 (3÷8)
2.
) 8x + 5 = 36x ^ 2
8x + 5 - 36x ^ 2 = 0 - 36x ^ 2 + 8x + 5 = 0
a × c = 36 × ( - 5) = - 180
b = - 8 - (36x ^ 2 + ( - 18 + 10) x - 5) = 0 - (36x ^ × - 18x + 10x - 5) = 0 - ((36x ^ 2 - 18x) + (10x - 5)) = 0 - (18x ( 2x - 1) + 5 ( 2x - 1)) = 0 - (2x - 1)(18x + 5) = 0
2x - 1 = 0
2x = 1
x = 1 ÷ 2
18x + 5 = 0
18x = - 5
x = - 5÷ 18
x = 1÷2 ; - 5÷18
x = 0, 5 ; - 0, 27
3.
) 27x ^ 2 + 12x - 7 = 0
a × c = 27 × ( - 7) = - 189
b = 12
27x ^ 2 + ( - 9 + 21)x - 7 = 0
27x ^ 2 - 9x + 21x - 7 = 0
(27x ^ 2 - 9x) + ( 21x - 7) = 0
9x (3x - 1) + 7(3x - 1) = 0
(3x - 1)( 9x + 7) = 0
3x - 1 = 0
3x = 1
x = 1÷3
9x + 7 = 0
9x = - 7
x = - 7÷9
x = 1÷3 ; - 7÷9
x = 0, 3 ; - 0, 7
4.
) 15x = 25x ^ 2 + 2 - 25x ^ 2 + 15x + 2 = 0
a × c = - 25 × - 2 = 50
b = 15 - 25x ^ 2 + (5 + 10) x - 2 = 0 - 25x ^ 2 + 5x + 10x - 2 = 0
( - 25x ^ 2 + 5x) + (10x - 2) = 0
5x ( - 5x + 1) - 2 ( - 5x + 1) = 0
( - 5x + 1) (5x - 2) = 0 - 5x + 1 = 0 - 5x = - 1
x = 1 ÷ 5
5x - 2 = 0
5x = 2
x = 2÷5
x = 1÷5 ; 2÷5
x = 0, 2 ; 0, 4
5.
) 8x ^ 2 - 2x - 3 = 0
a×c = 8 × - 3 = - 24
b = - 2
8x ^ 2 + ( - 6 + 4) x - 3 = 0
8x ^ 2 - 6x + 4x - 3 = 0
(8x ^ 2 - 6x) + (4x - 3) = 0
2x (4x - 3) + 1( 4x - 3) = 0
(4x - 3) (2x + 1) = 0
4x - 3 = 0
4x = 3
x = 3÷4
2x + 1 = 0
2x = - 1
x = - 1÷2
x = 3÷4 ; - 1÷2
x = 0, 75 ; - 0, 5
6.
) 105 = x + 2x ^ 2 - 2x ^ 2 - x + 105 = 0
a × c = 2 × - 105 = - 210
b = 1
2x ^ 2 + ( - 14 + 15)x - 105 = 0
2x ^ 2 - 14x + 15x - 105 = 0
(2x ^ 2 - 14x) + (15x - 105) = 0
2x(x - 7) + 15 (x - 7) = 0
(x - 7) (2x + 15) = 0
x - 7 = 0
x = 0 + 7
x = 7
2x + 15 = 0
2x = - 15
x = - 15 ÷ 2
x = 7 ; - 15 ÷2
x = 7 ; - 7 (1÷2) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.
) x ^ 2 - 3x + 2 = 0
x ^ 2 + bx + c = 0 - 2 ; - 1 = 0
(x - 2) (x - 1) = 0
x - 2 = 0
x = 0 + 2
x = 2
x - 1 = 0
x = 0 + 1
x = 1
x = 2, 1
2.
) x ^ 2 - 2x - 15 = 0
x ^ 2 + bx + c = 0 - 5 ; 3 = 0
(x - 5) (x + 3) = 0
x - 5 = 0
x = 0 + 5
x = 5
x + 3 = 0
x = 0 - 3
x = - 3
x = 5 ; - 3
3.
) x ^ 2 - 19x = 88
x ^ 2 + bx + c = 0
x ^ 2 - 19x - 88 = 0 - b + - raiz b ^ 2 - 4ac ÷ 2a
a = 1
b = - 19
c = - 88
19 + - raiz ( - 19) ^ 2 - 4(1)( - 88) ÷ 2(1)
19 + - raiz 361 + 352 ÷ 2
x = 22, 8 ; - 3, 85
4.
) x ^ 2 + 4x = 285
x ^ 2 + bx + c = 0 - 15 ; 19 = 0
(x - 15) (x + 19) = 0
x - 15 = 0
x = 0 + 15
x = 15
x + 19 = 0
x = 0 - 19
x = - 19
x = 15, - 19
5.
) 5x(1 - x) - 2(2x ^ 2 - 7x) = - 8
5x × 1 + 5x ( - x) - 2 ( 2x ^ 2 - 7x) = - 8
5x + 5x ( - x) - 2 (2x ^ 2 - 7x) = - 8
5x + 5 (xx) × - 1 - 2 (2x - 7x)
a ^ m a ^ n = a ^ m + n
5x - 5x ^ 2 - 2 (2x ^ 2 - 7x) = - 8
5x - 5x ^ 2 + ( - 4x ^ 2 + 14x) = - 8 - 5x ^ 2 - 4x ^ 2 + 19x = - 8 - 9x ^ 2 + 19 x = - 8 - 9x ^ 2 + 19x + 8 = 0 - b + - raiz b ^ 2 - 4ac ÷ 2a - ( - 19) + - raiz ( - 19) ^ 2 - 4 (9) ( - 8) ÷ 2 (9)
x = 19 + - raiz 649 ÷ 18
x = 2, 4 ; - 0, 35
6.
)x ^ 2 - (7x + 6) = x + 56
x ^ 2 - 7x - 6 = x + 56
x ^ 2 - 7x - 6 - x = 56
x ^ 2 - 8x - 6 = 56
x ^ 2 - 8x - 6 - 56 = 0
x ^ 2 - 8x - 62 = 0 - b + - raiz b ^ 2 - 4ac ÷ 2a
a = 1 ; b = - 8 ; c = - 62 - ( - 8) + - raiz ( - 8) ^ 2 - 4 (1)( - 62) ÷ 2(1)
8 + - raiz 64 + 248 ÷ 2
x = 8 + - raiz 312 ÷ 2
x = 4 + - raiz 78
x = 12, 8 ; - 4, 83
7.
) (x - 1) ^ 2 + 11x + 199 = 3x ^ 2 - (x - 2) ^ 2
(x - 1) ^ 2 + 11x + 199 - 3x ^ 2 + (x - 2) ^ 2 = 0
xx + x × - 1 - 1x - 1 × - 1 + 11x + 199 - 3x ^ 2 + ( x - 2) ^ 2 = 0
x ^ 2 - 2x + 1 + 11x + 199 - 3x ^ 2 + ( x - 2)(x - 2) = 0
x ^ 2 - 2x + 1 + 11x + 199 - 3x ^ 2 + ( x ^ 2 - 4x + 4x) = 0 - x ^ 2 + 9x + 1 + 199 - 4x + 4 = 0 - x ^ 2 + 5x + 204 = 0 - 17 ; 2 = 0 - ( x - 17) (x + 12) = 0
x - 17 = 0
x = 0 + 17
x = 17
x + 12 = 0
x = 0 - 12
x = - 12
x = 17 ; - 12
8.
) (x - 2)(x + 2) - 7(x - 1) = 21
x ^ 2 - 4 - 7(x - 1) = 21
x ^ 2 - 7x + 3 = 21
x ^ 2 - 7x + 3 - 21 = 0
x ^ 2 - 7x - 18 = 0 - 9 ; 2 = 0
(x - 9)(x + 2) = 0
x - 9 = 0
x = 0 + 9
x = 9
x + 2 = 0
x = 0 - 2
x = - 2
x = 9, - 2
9.
)2x ^ 2 - (x - 2)(x + 5) = 7(x + 3)
2x ^ 2 + ( - x ^ 2 - 3x + 10) = 7(x + 3)
2x ^ 2 - x ^ 2 - 3x + 10 = 7 ( x + 3)
x ^ 2 - 3x + 10 = 7 ( x + 3)
x ^ 2 - 3x + 10 = 7x + 21
x ^ 2 - 3x + 10 - 7x = 21
x ^ 2 - 10x + 10 = 21
x ^ 2 - 10x + 10 - 21 = 0
x ^ 2 - 10x - 11 = 0 - 11 ; 1 = 0
(x - 11) ( x + 1) = 0
x - 11 = 0
x = 0 + 11
x = 11
x + 1 = 0
x = 0 - 1
x = - 1
x = 11 ; - 1
10.
)(x - 1)(x + 2) - (2x - 3) (x + 4) - 4 + 14 = 0
x ^ 2 + x - 2 - (2x - 3) (x + 4) - 4 + 14 = 0
x ^ 2 + x - 2 + (2x + 3) (x + 4) - 4 + 14 = 0
x ^ 2 + x - 2 + ( - 2xx - 2x × 4 + 3x + 3 × 4) - 4 + 14 = 0
x ^ 2 + x - 2 + ( - 2x ^ 2 - 5x + 12) - 4 + 14 = 0 - x ^ 2 - 4x - 2 + 12 - 4 + 14 = 0 - x ^ 2 - 4x + 20 = 0
x ^ 2 + 4x - 20 = 0 - b + - raiz b ^ 2 - 4ac ÷ 2a
a = 1
b = 4
c = - 20 - 4 + - raiz ( - 4) ^ 2 - 4(1)( - 20) ÷ 2(1)
x = - 4 + - raiz 16 + 80 ÷ 2
x = - 4 + - raiz 96 ÷ 2
x = - 4 + - raiz 4 ^ 2 × 6
x = - 4 + - 4 raiz 6 ÷ 2
x = - 2 + - 2 raiz 6
x = 2, 89 ; - 6, 89.