1) x² - 2x - 48 = 0;
D = b² - 4ac;
D = (-2)²- 4 • 1 • (-48) = 4 + 192 = 196; √D = 14;
x = (-b ± √D)/(2a);
x1 = (2 + 14)/2 = 16/2 = 8;
x2 = (2 - 14)/2 = - 12/2 = -6;
x² - 2x - 48 (x - 8)(x + 6).
2) 2x²- 5x + 3 = 0;
D = (-5)² - 4 • 2 • 3 = 25 + 24 = 49; √D = 7;
x1 = (5 + 7)/4 = 12/4 = 3;
x2 = (5 - 7)/4 = -2/4 = -0,5;
2x² - 5x + 3 = 2(x + 0,5)(x - 3) = (2x + 1)(x - 3).
3) 3x² - 10x + 3 = 0;
D = (-10)² - 4 • 3 • 3 = 100 - 36 = 64; √D = 8;
x1 = (10 + 8)/6 = 18/6 = 3;
x2 = (10 - 8)/6 = 2/6 = 1/3;
3x² - 10x + 3 = 3(x - 1/3)(x - 3) = (3x - 1)(x - 3).
4) 5x² - x - 42 = 0;
D = (-1)^2 - 4 • 5 • (-42) = 1 + 840 = 841; √D = 29;
x1 = (1 + 29)/10 = 30/10 = 3;
x2 = (1 - 29)/10 = -28/10 = -2,8;
5x² - x - 42 = 5(x + 2,8)(x - 3) = (5x +14)(x - 3).
5) 3x² - 8x + 5 = 0;
D = (-8)^2 - 4 • 3 • 5 = 64 - 60 = 4; √D = 2;
x1 = (8 + 2)/6 = 10/6 = 5/3;
x2 = (8 - 2)/6 = 6/6 = 1;
3x² -8x + 5 = 3(x - 5/3)(x - 1) = (3x - 5)(x - 1).
6) 36x² - 12x + 1 = 0;
D = (-12)^2 - 4 • 36 • 1 = 144 - 144 = 0;
x1 = x2 = 12/72 = 1/6;
36x²- 12x + 1 = 36(x - 1/6)(x - 1/6) = 6(x - 1/6) * 6(x - 1/6) = (6x - 1)(6x - 1).
1) x² - 2x - 48 = 0;
D = b² - 4ac;
D = (-2)²- 4 • 1 • (-48) = 4 + 192 = 196; √D = 14;
x = (-b ± √D)/(2a);
x1 = (2 + 14)/2 = 16/2 = 8;
x2 = (2 - 14)/2 = - 12/2 = -6;
x² - 2x - 48 (x - 8)(x + 6).
2) 2x²- 5x + 3 = 0;
D = (-5)² - 4 • 2 • 3 = 25 + 24 = 49; √D = 7;
x1 = (5 + 7)/4 = 12/4 = 3;
x2 = (5 - 7)/4 = -2/4 = -0,5;
2x² - 5x + 3 = 2(x + 0,5)(x - 3) = (2x + 1)(x - 3).
3) 3x² - 10x + 3 = 0;
D = (-10)² - 4 • 3 • 3 = 100 - 36 = 64; √D = 8;
x1 = (10 + 8)/6 = 18/6 = 3;
x2 = (10 - 8)/6 = 2/6 = 1/3;
3x² - 10x + 3 = 3(x - 1/3)(x - 3) = (3x - 1)(x - 3).
4) 5x² - x - 42 = 0;
D = (-1)^2 - 4 • 5 • (-42) = 1 + 840 = 841; √D = 29;
x1 = (1 + 29)/10 = 30/10 = 3;
x2 = (1 - 29)/10 = -28/10 = -2,8;
5x² - x - 42 = 5(x + 2,8)(x - 3) = (5x +14)(x - 3).
5) 3x² - 8x + 5 = 0;
D = (-8)^2 - 4 • 3 • 5 = 64 - 60 = 4; √D = 2;
x1 = (8 + 2)/6 = 10/6 = 5/3;
x2 = (8 - 2)/6 = 6/6 = 1;
3x² -8x + 5 = 3(x - 5/3)(x - 1) = (3x - 5)(x - 1).
6) 36x² - 12x + 1 = 0;
D = (-12)^2 - 4 • 36 • 1 = 144 - 144 = 0;
x1 = x2 = 12/72 = 1/6;
36x²- 12x + 1 = 36(x - 1/6)(x - 1/6) = 6(x - 1/6) * 6(x - 1/6) = (6x - 1)(6x - 1).
1) 2cosx-1 < 0
cosx < 1/2
arccos(1/2) + 2πn < x < 2π - arccos(1/2) + 2πn, n ∈ Z
π/3 + 2πn < x < 2π - π/3 + 2πn, n ∈ Z
π/3 + 2πn < x < 5π/3 + 2πn, n ∈ Z
2) sin2x - √2/2 < 0
sin2x < √2/2
- π - arcsin(√2/2) + 2πk < 2x < arcsin(√2/2) + 2πk, k ∈ Z
- π - π/4 + 2πk < 2x < π/4 + 2πk, k ∈ Z
- 5π/4 + 2πk < 2x < π/4 + 2πk, k ∈ Z
- 5π/8 + πk < x < π/8 + πk, k ∈ Z
3) tgx<1
- π/2 + πn < x < arctg(1) + πn, n ∈ Z
- π/2 + πn < x < π/4 + πn, n ∈ Z