x ,
a, b, c, ,
.
.
1. : .
2 + 10 - 24 = 0. :
2 + 10 - 24 = 2 + 12 - 2 - 24 = ( + 12) - 2( + 12) = ( + 12)( - 2).
, :
( + 12)( - 2) = 0
, , , . = 2, = - 12. , 2 - 12 2 + 10 - 24 = 0.
2. : .
2 + 6 - 7 = 0. .
2 + 6 :
2 + 6 = 2 + 2 3.
- , - 3. , 32,
2 + 2 3 + 32 = ( + 3)2.
2 + 6 - 7 = 0,
32. :
2 + 6 - 7 = 2 + 2 3 + 32 - 32 - 7 = ( + 3)2 - 9 - 7 = ( + 3)2 - 16.
, :
( + 3)2 - 16 =0, ( + 3)2 = 16.
, + 3 - 4 = 0, 1 = 1, + 3 = -4, 2 = -7.
3. : .
2 + b + = 0, ≠ 0
4 :
422 + 4b + 4 = 0,
((2)2 + 2 b + b2) - b2 + 4ac = 0,
(2ax + b)2 = b2 - 4ac,
2ax + b = √ b2 - 4ac,
2ax = - b √ b2 - 4ac,
.
) : 42 + 7 + 3 = 0.
= 4, b = 7, = 3, D = b2 - 4ac = 72 - 4 4 3 = 49 - 48 = 1,
D > 0, ;
, , ..
b2 - 4ac >0, 2 + b + = 0 .
) : 42 - 4 + 1 = 0,
= 4, b = - 4, = 1, D = b2 - 4ac = (-4)2 - 4 4 1= 16 - 16 = 0,
D = 0, ;
, , .. b2 - 4ac = 0,
2 + b + = 0 ,
) : 22 + 3 + 4 = 0,
= 2, b = 3, = 4, D = b2 - 4ac = 32 - 4 2 4 = 9 - 32 = - 13, D < 0.
.
, , .. b2 - 4ac < 0,
2 + b + = 0 .
(1) 2 + b + = 0 ( ), . (1) : , , , , .
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4. : .
,
2 + px + c = 0. (1)
, =1
x1 x2 = q,
x1 + x2 = - p
( p q ).
) q (1) (q > 0), p. < 0, , < 0, .
,
x2 3x + 2 = 0; x1 = 2 x2 = 1, q = 2 > 0 p = - 3 < 0;
x2 + 8x + 7 = 0; x1 = - 7 x2 = - 1, q = 7 > 0 p= 8 > 0.
) q (1) (q < 0), , , p < 0, , p > 0.
,
x2 + 4x 5 = 0; x1 = - 5 x2 = 1, q= - 5 < 0 p = 4 > 0;
x2 8x 9 = 0; x1 = 9 x2 = - 1, q = - 9 < 0 p = - 8 < 0.
.
1) 3452 137 208 = 0.
. + b + = 0 (345 137 208 = 0),
1 = 1, 2 = c/a = -208/345.
: 1; -208/345.
2) 1322 247 + 115 = 0.
. + b + = 0 (132 247 + 115 = 0),
1 = 1, 2 = c/a = 115/132.
: 1; 115/132.
. b = 2k ,
.
32 14 + 16 = 0.
. : = 3, b = 14, = 16, k = 7;
D = k2 ac = (- 7)2 3 16 = 49 48 = 1, D > 0, ;
: 2; 8/3
.
2 + + q= 0
, = 1, b = = q.
:
(3) , .
. 2 14 15 = 0.
. : 1,2 =7
: 1 = 15; 2 = -1.
5. : .
. 2 - 2 - 3 = 0.
.
= 2 - 2 - 3
1) : = 1, b = -2, 0 = = 1, 0 = f(1)= 12 - 2 - 3= -4. , (1; -4), = 1.
2) , , = -1 = 3.
f(-1) = f(3) = 0. (-1; 0) (3; 0).
3) (-1; 0), (1; -4), (3; 0) (. 68).
2 - 2 - 3 = 0 ; , : 1 = - 1, 2 3.