c
a c2 = a2 + b2
b a<1
a2 > a
a = 1/2
1/4
1/2
(1/2)2 =
P P
Q
Q
P c
a
Q c2 = a2 + b2 P P P P
P
Q
Q
P
Q P = Q
⇒Q a<1
a2 > a
b
• •
A B A B d1 , e1 , v1 A B B
d 2 , e2 , v 2 A
v1 > v2
d1 + e1 + v1 = d2 + e2 + v2 . A e1 + 3 v 1 B
e2 + 3v2 e1 + 3v1 = e2 + 3v2 .
3(v1
−v )=e −e 2
2
−
−d
2
= e2
2
1
1
e2 e1 > 0
− e − e + (v − v ) = e − e − 3 d −d > 0 e −e > 0 e2
1
1
A
1
− v = − e −3 e .
−
v1 v2 > 0
d1
v2
2
1
2
2
1
2
B
1
1
2 = ( e2 3
− e ). 1
P
⇒Q
P = Q
Q P
a
• • •
b
c
T c2 = a2 + b2
a
b
b
β a
γ Q
a
α b
c
a+b a Q c a γ = 180
−
b (α + β ) = 180◦
− 90◦ = 90◦
(a + b)2 = 4
· ab2 + c , 2
a2 + 2ab + b2 = 2ab + c2 , a2 + b2 = c2 ,
b
N 2
• •
N
N 2 N
N 2
N 2 N
N 2
N 2
N
• •
N N 2 N p
N 2 = (2 p + 1)(2 p + 1)
= 4 p2 + 2 p + 2 p + 1 = 4 p2 + 4 p + 1 = 2(2 p2 + 2 p) + 1 = 2q + 1, q = 2 p2 + 2 p
N 2 = 2q + 1
N N = 2 p +1
• • •
x
• •
x + 1/x
≥2
x x + 1/x
x+
1
x
≥2 x <2
x>0
x2 + 1 < 2x.
x2
−
x2 2x + 1 < 0 2x + 1 = (x 1)2
−
−
(x
2
− 1)
<0 x + 1/x
≥2
9
• • • • • • • • •
• •
• •
0 xyz = 36 y
z x y
1 1 1 1 1 2 2 3
x y
1 1 1 1
z
1 36 2 18 3 12 4 9 6 6 2 9 3 6 3 4
z
1 36 2 18 3 12 4 9
1 6
6
2 2 2 3 3 3
9 6 4
xyz
36 36 36 36 36 36 36 36
x+y+z
38 21 16 14
11 10
x
y
z
36 x
A B
A B C
A
A B C
C
A
A C A A
D B
B
C A
B
D
B A C D C A
D
B
B
◦
◦
◦
◦
◦ ◦
◦
◦
◦ ◦
1, 2, . . . , 9,
i
j i
j
• •
•
•
2
3
4
5
6
7
8
9
•
•
• •
1
• •
•
•
•
• •
•
• •
Q
• • • •
T
3
4
6
•
...
1, 2, 3, . . . , 99
2, . . . , 20
x
x
3x
− 3x = 2x − 2x.
3(x
− x) = 2(x − x).
−x
x
−x
3=2
x y z
x
x 2x
x/5
2x +
x = 121, 5
10x + x = 605, 11x = 605
x = 605/11 = 55
x ax + b = 0,
a=0 b
2x
∈R
x
−3=0
−4x + 1 = 0 3 x − π = 0. 2
a b
a+c a=b
⇒
=
a + c = b + c.
c x
5x
− 3 = 6,
b+c
x
3 (5x
− 3) + 3 = 6 + 3
5x = 9.
a b
a c
·
a=b
⇒
=
b c
·
ac = bc.
5x = 9 1/5 5x 9 x= = , 5 5 5x
−3=6
x
• • • • • •
x
6x 6x + 12 6x + 12 = 2x + 4 3 2x + 4
1=2 x + 2x = 2x + x. x
(x
− x) 1(x (x
1=2
− x = 2x − 2x
− x)
− x) = 2(x − x)
− 2x = 4
ax + b = 0
−b
•
ax + b + ( b) = 0 + ( b)
− ⇐⇒ ax = −b.
−
•
a=0 x ax b = a a
− ⇐⇒ x = − b . a
ax + b = 0 b x=− . a
x x11 + 11x + 1x1 = 777, x
abc
10 102 a + 10b + c x11 = 100x + 11 11x = 110 + x 1x1 = 101 + 10x
100x + 11 + 110 + x + 101 + 10x = 777
x=
111x + 222 = 777
−
777 222 = 5. 111
1
1
9
6 9
1
s
9
45 = 1 + 2 + s
15
··· + 9 = 3s, x
x
1 + x + 6 = 15 8 9
x=8
15
9 x
x x
15
·
x
4 15 = 60 x
x
1+2+3+4+ 45 + 3x = 60
··· + 9 + 3x = 60, x=5
6.378.000
C
r C = 2πr,
π 3, 1416 C T
∼ × 3, 1415 × 6.378.000 = 40.072.974
C T = 2πr T = 2
rT x
rf x = rf
rf
40.072.975 = 2πr f
rf =
−r
T
C T + 1 = 40.072.975
40.072.975 . 2π
rf
6.378.000, 16 rf rT = 0, 16
x
x =
−
π π
π
π
π∼ = 3, 1415926535897932384626433832795.
x y
2x + 3y = 14.
x + y = 6.
x1 x2 . . . xn a1 x1 + a2 x2 +
··· + a x
n n
+ b = 0,
a1 , a2 , . . . , an
b
2x
− 3y = 0 x c b+ =5 3
−
2a
y
a, b
c.
(r1 , r2 , . . . , rn ) r1 x2 r2 xn rn a1 r1 + a2 r2 + + an rn + b = 0. 2x 3y = 0
x1
··· −
(3, 2)
· − 3 · 2 = 0.
2 3 2x (2, 0, 3)
− 3y = 0
· −
2 2
· − · −
(2, 3) 5 = 0.
−
2 2 3 3= c 2a b + = 5 3
3 0 + = 5. 3 n
x1 x2 . . . xn
k x1 , x2 , . . . , xn
··· ··· ··································· a11 x1 + a12 x2 +
+ a1n xn + b1 = 0,
a21 x1 + a22 x2 +
+ a2n xn + b2 = 0,
ak1 x1 + ak2 x2 +
··· + a
kn
xn + bk = 0,
aij (1
≤ i ≤ k, 1 ≤ j ≤ n) x j
(r1 , r2 , . . . , rn ) (r1 , r2 , . . . , rn )
2x + 3y = 14, x + y = 6.
x x=6
− y.
aij
− y) + 3y = 14, 12 − 2y + 3y = 14,
2(6
y = 2.
y=2
x y
x=6
z
−
−2=4
x+y x
− z − 1 = 0, y − 1 = 0. x = y + 1.
(y + 1) + y
− z − 1 = 0, 2y − z = 0,
z = 2y.
x
z
y y
t t
x
z t x = t + 1,
y = t,
z = 2t
x y
z
x + y + 2z
− 2 = 0, y + z − 3 = 0. x+z
x=2
(2
− 1 = 0,
−z
y=3
− z.
− z) + (3 − z) + 2z − 1 = 0 ⇐⇒ 4 = 0,
p
g
2g = p
3(g
− 1.
− 1) = p. 2g = 3g
g=4
p=9 A1 V 1
V 2 A2 A1
A1 + A2 = π4 , A1 + 2A2 = 1,
A2
−3−1
A1
A2 π A1 = 4
π 4
A2 = 1
−
−A
2
−A ; 2
+ 2A2 = 1,
π + A2 = 1. 4 A1 = π4 1 π4 =
π
− −
4
x y
π 2
−1
y
15x + 15y = c. 20 20x = c.
15x + 15y = 20x
15y = 20x
3y = x
5 20
x
60y = 20
×3
−y
30
×
× 3y = 20x = c
3y y = 2y 2y = 60y = c
−
− 15x = 5x.
30
x2
− 6x − 8 = 0.
x2
− 6x = 8. x2
9 (x
2
− 3)
x1
2
− 3) = 9 + 8 = 17. √ x − 3 = − 17 √ √ x = 3 − 17. = 3 + 17 (x
√ x − 3 = 17
− 6x + 9
2
a, b
ax2 + bx + c = 0,
a = 0 b, c
∈R
x
x
ax2 + bx =
−c
c
a
−
b c x2 + x = . a a
b2 4a2
2
b x+ 2a
b 2 = x +2 x+ 2a
2
b 2a
b2 = 2 4a
−
c b2 4ac . = a 4a2
−
b2 4ac
−
b x+ 2a
2
=
≥0 b x+ 2a
√b − 4ac =
x1 = −
2
4a2
b x+ 2a
2a
b + 2a
− 4ac =
√
∆ . 4a2
≥0
∆ ∆
b2
∆
2
b
√b − 4ac . =− 2
2a
− 4ac = −b +
2a
√
2a
∆
x2 = −
• • •
b 2a
√
2
b
−
− 4ac = −b − 2
4a
√
2a
∆
.
∆>0 x1 = x2 =
∆=0
−b/2a
∆<0
2x2
− 4x + 2 = 0.
−4 c = 2 − 4ac = (−4) − 4 · 2 · 2 = 0. b 4 x=− = =1 2a 4
a = 2, b =
∆ = b2
2
x2
−1 ∆ − 4ac = (−1) − 4 · 1 · (−1) = 5.
a=1 b=
∆ = b2
x1 =
√ −b + ∆ 2a
− x − 1 = 0.
−1
c=
2
=
√ 1+ 5 2
x2
√ √ − b− ∆ 1− 5 . = = 2a
2
x x = 1+
1 1 1+ x
,
x
1+ 1
1+
x+1 1 . = x x
= 1+
1 x 1 + 2x x= 1+x 1+
x2 + x = 1 + 2x
x1 =
√ 1+ 5 2 (
x 1 + 2x . = 1+x 1+x
2
⇐⇒ x − x − 1 = 0. x2 =
√ + )/
√ 1− 5 2
.
an n
≥0
an+2 = an+1 + an . an = xn
an
x=0 an
an
xn
n xn+2
n+1
−x −x
n
≥0
= 0.
xn xn (x2
− x − 1) = 0 xn = 0 x2 x 1 = 0 x2 x 1 = 0
xn = 0
x=0
an =
√ 1+ 5 2
− −
n
an bn α β βbn an = xn1
bn = xn2
an =
− −
−√ 1
5
2
n
.
αan +
a, b
ax2 + bx + c = 0 x1 x2 x1 x2
c
x1 =
≥0 √ −b − ∆
∆
√ −b + ∆
x2 =
2a x2
x1
x1 + x2
a=0
2a
.
√ √ − − b + ∆ −b − ∆ b 2b − . = + = = 2a
2a
2a
a
x1 x2
x1 x2 =
= =
√ b+ ∆
− −
− − √ · √ − − √ b
2a b+
∆
2a
c 4ac . = 2 a 4a
∆ 4a2
b
∆
a=1 α x2
β
− sx + p = 0
α+β =s
αβ = p.
b2 ∆ = 4a2
−
α
(x α
β
2
− α)(x − β) = x − sx + p
β ax2 + bx + c = 0 sx + p) = 0 s = b/a sx + p = 0 α β
a(x2 x2
− −
a=0 p = c/a
ax2 + bx + c = a(x2
−
− sx + p) = a(x − α)(x − β )
α
β
ax2 + bx + c = 0 α
(x
− α) m2
a
a
b
a + b = 12, ab = 28.
b
∆ = 122
x2 12x + 28 = 0 4 28 = 32
−·
−
√ 12 + 32
a=
b=
2
√ 12 − 32 2
√
=6+2 2
=6
√
−2
2.
x2 + bx + 17 = 0 b
m m= n b m n m=1 n=1 1 + b + 17 = 0 b = 18 b
n n+m =
m=n
− −
−b
·
m n = 17
−
a
(i, j )
P = {1, 2, . . . , n}
j
1
A1 = (1, 2), (1, 3), . . . , (1, n) .
i
2 1
A2 = (2, 3), (2, 4), . . . , (2, n) .
(2, 1)
(1, 2)
1
2
Ai = (i, i + 1), (i, i + 2), . . . , (i, n) , Ai
∩A
j
=
i = j
∅
1
≤ i ≤ n − 1.
Ai
A1
|X |
X a
∪ ··· ∪ A −
n 1
|(A ∪ A ∪ · · · ∪ A − )| = |A | + |A | + ··· + |A − | = (n − 1) + (n − 2) + ··· + 2 + 1 (n − 1)n = = a. 1
2
n 1
1
2
n 1
2
n2
− n − 2a = 0
a < 75
n n1
= n1 11 12 = 132
−n n ≥ · 2
1
−n n 2
1
−n
= 2a
2
n2
≥ 12
n1
≥ 12
−
n1 n2 = 2a, n1 + n2 = 1,
− 1 ≥ 11 −n ≥ 11 2
≥ 12 · 13 = 156
n1
≥ 12 a
≥
n1 78
≥ 13
a < 75 n1 = 12 a = 66
x4
− 2x
2
+ 1 = 0.
y x2 y = x2
(y
− 1)
2
x=
y=1
0 = y2 x2 = y = 1
x=1
−1 ax2k + bxk + c = 0,
− 2y + 1 =
k
y = xk ay 2 + by + c = 0,
∈ N,
y = α α
xk = α,
• • •
x=
√α
α<0 x=
k
k
± √α k
k α>0
k
ax2 + bx + c = 0
Ay 2 + B = 0, A B
y1 =
a,b,c
−
B A
y2 =
− −
B , A
B A
− ≥ 0.
x = u+v u v a(u + v )2 + b(u + v ) + c = 0,
au2 + 2auv + av 2 + bu + bv + c = 0.
v av 2 + (2au + b)v + au2 + bu + c = 0.
u
(2au + b)v
−b/2a − b b − b 2a + c = 0 ⇐⇒ av +a 2a
u=
2
av 2
av
v1 =
− b
2
4ac 4a2 u=
2
−b +
v2 =
−
−b/2a
x1 =
−
b + v1 2a
2
2 b 2 + 4a
−
b2 + c = 0, 2a
+ 4ac = 0. 4a
− b2
4ac
4a2
,
∆ = b2
x=u+v
x2 =
−
b + v2 , 2a
− 4ac ≥ 0.
x x
3x
≤ 100. ·
99 = 3 33 3 34 = 102 > 100
·
x = 33
x x
3x 100 < 0 100/3 3x
−
− 100 < 0
a, b
∈R
ax + b < 0, ax + b > 0, ax + b
≤ 0,
ax + b
≥ 0,
a = 0.
S
•
a
b
a c <,
•
≥
•
a+c
b
a
≤ bc
c <,
a b bc
≤ b+c
>
a ac
≤b
≥ a
c <,
≥
≤b
>
≤b >
ac
≥
ax + b < 0
•
ax + b > 0.
a>0 a
ax + b < 0
x + b/a < 0
−b/a
−b/a = {x ∈ R; x < −b/a}, x<
•
−b
a
ax + b > 0
= x
{ ∈ R; x > −b/a}, •
−b
•
a
a<0 ax + b < 0
x>
−b/a = {x ∈ R; x > −b/a},
a x + b/a > 0
•
−b
a
ax + b > 0
{ ∈ R; x < −b/a},
= x
•
−b ax + b
≤0
ax + b
a
≥0
x=
−b/a − ≥0
8x 4
x
− 4/8 + 1/2 ≥
1/2 1/2,
{ ∈ R; x ≥ 1/2}.
= x
a = 3456784 3456786 + 3456785 b = 34567852
·
− 3456788
x 3456784 a b a = x (x + 2) + (x + 1) b = (x + 1)2 a = x2 + 3x + 1 b = x2 + x 3 a
·
x2
− + 3x + 1 ≤ x
2
+x
− (x + 4) ≤b
− 3,
2
−x −x+3 2x + 4
≤ 0. x
x
≤ −2,
x = 3456784 b
a
∈R
a>b
3
×3
1+2+3+ 45
x
· ·· + 9 =
x
•
x
17 > 15
•
x y
1 + 2 + y = 15
⇔y
= 12 x
z z = 15 x
− (x + 9) ≥ 1 ⇔ 6 − x ≥ 1,
≤5
x s
− ≤ ≤
s = 15 (x + 1) = 14 5 x 5
−x ≤ 9
x
x=5
≥5 a b P
c x
y
a>b
c
P
= x+y
P
= x+y
C a x P c
B
b
y
A P
AC
BC
S a b
y ax by + = S, 2 2 ax = 2S by 2S by x= . a
− −
y
− by + y a 2S − by + ay = a 2S a − b y, = +
x+y =
2S
a
a
x
= α + βy,
α=
0
2S a hb
≤y≤
β=
a
− b. a
hb
b a > b α + βhb
0
β
≤ βy ≤ βh
0
α
b
≤ ≤ α + βh . b
A
P y = hb
≤ α + βy ≤
y =0
P
B
ax2 + bx + c
a,b,c
∈R
ax2 + bx + c < 0, ax2 + bx + c > 0, ax2 + bx + c
≤ 0,
ax2 + bx + c
≥ 0,
a = 0.
a ax2 + bx + c
x2 3x + 2 > 0 x2 3x + 2 = 0
− −
x2
− 3x + 2 = (x − 1)(x − 2). (x (x
•
− 1)
(x
x
−1>0⇔x>1
x
− 2 > 0 ⇔ x > 2,
x
−1<0⇔x<1
x
− 2 < 0 ⇔ x < 2,
− 2)
− 1)(x − 2)
x>2
•
x<1 x2
− 3x + 2 > 0
x<1
ax2 + bx + c > 0.
x>2
− − − −
b ax2 + bx + c = a x2 + x + a b 2 =a x + x+ a b = a x2 + x + a b =a x+ 2a
∆ = b2
c a b2 4a2 b2 4a2
2
b2 c + 4a2 a b2 a 4a2
∆ , 4a
− 4ac. ∆ = b2
− 4ac > 0
a
•
a> 0 2
−
b a x+ 2a a>0
∆ > 0. 4a
1/a
2
− b x+ 2a
∆>0
∆ > 0. 4a2
c a
2
− x+
b 2a
∆ = 4a2 =
− √ √ − √ − − √ − √ 2
x+
b 2a
x+
b+ ∆ 2a
2
∆ 2a
x+
b
2a
− b 2a ∆ x − = (x − α)(x − β ) > 0, =
√ b− ∆ − α = 2a
(x (x
− β)
α<β x<α
x
√ b+ ∆ − β = 2a
− α)(x − β) > 0
(x x > β
x<α
α<β
{ ∈ R; x < α
}
= x
•
x>β ,
•
α
•
b+ ∆ 2a
ax2 + bx + c = 0
x > α x>β x<β
∆
β
a< 0
1/a 2
− b x+ 2a
∆ < 0, 4a2
− α)
(x √ b− ∆ − α = 2a
x
x
−
−
− α)(x − β) < 0,
√ b+ ∆ − β = 2a
ax2 + bx + c = 0
(x α) (x β ) α>0 x β <0 x αβ a<0 β > 0 β < x < α
−
−
−
{ ∈ R; β < x < α},
= x
•
•
α
β
∆ = b2
− 4ac = 0 2
b a x+ 2a x =
−
a<0
∆ = b2 x
> 0, b 2a
a > 0
− 4ac < 0
a 2
−
b 2 ax + bx + c = a x + 2a
∆ > 0, 4a
x
−α <0
−
∆ 4a
>0
a 2
−
b 2 ax + bx + c = a x + 2a
−
∆ 4a
∆ 4a
<0
ax2 + bx + c < 0
ax2 + bx + c
ax2 + bx + c
≥0
≤0 α β
−b/2a
x>0 x+
x
∈R
(x x2
1 x
≥ 2. 2
− 1) ≥ 0 2
− 2x + 1 ≥ 0 ⇐⇒ x
+1
≥ 2x.
x
1 x+ x
≥ 2,
f (x) = ax2 + bx + c 2
−
b ax2 + bx + c = a x + 2a
∆ = b2 f (x)
∆ , 4a
− 4ac.
f (x)
x a > 0 x = a < 0 x=
−
b 2a
−
b
−
f (
−
f (
2a
b 2a
)=
−
2
−a
) =
−
∆ 4a
∆ 4a
a, b ab
b 2a
a+b = 1
≤ 1/4 ab = a(1 f (a)
+a
f (a) =
− a) = −a ≤ 1/4
2
+a
f (a) = 0 < a < 1
f (a)
2
2
−(a − a) = −(a − a + 1/4 − 1/4) = −(a − 1/2) 1/4
2
+ 1/4,
a = 1/2
ABCD r ABCD D
C r y x
A
B
A = 2x 2y = 4xy.
·
y=
√r
2
√
2
−x ,
A = 4x r 2
2
−x . ABCD A2
A2 A2 = 16x2 (r2
2
2
2
4
− x ) = 16r x − 16x .
z = x2 2
A =
−16z
2
2
+ 16r z =
−16
− z
A2 x=
2
y=
2r 2
√
√ =r 2
|2x − 5| = 3 |2x − 3| = 1 − 3x
r
2
−
2
2
+ 4r 4 , z =
r √
r2
r2 r . = 2 2
√
r2 2
|3 − x| − |x + 1| = 4
|2x − 5| = 3.
|a| =
−
|a| = b ⇐⇒
a
a
≥ 0,
a
a < 0. b
a=b
a=
−b.
x
x
2x
−5=3
2x
− 5 = −3. x1 = 4
x2 = 1
|2x − 3| = 1 − 3x
(a)
2x
− 3 ≥ 0, 2x − 3 = 1 − 3x,
(b)
2x 3 = 1 3x 2x 3 0 (2x 3) = 1 3x 2x 3 < 0
− − −
− − ⇔ 5x = 4 − ≥ − − − −
− ≥ − 1
3x
(2x
x1 = 4/5 1 3x1 0
−
≥
|x + 1|
5x2 + 6x x2 = 2
−
2x
3 < 0,
(2x
3) = 1
− 3x.
x = 4/5 x=
−2
x=
−2
0,
3)2 = (1
2
− 3x) . (2x
−8 = 0
2
− 3)
= (1
2
− 3x)
x1
−2 |3 − x| − |x + 1| = 4 x2 =
x1 = 3
x2 =
−1
|3 − x|
−1 −1 ≤ x ≤ 3
x < x<
x> 3
−1 3
− x − (−x − 1) = 4 ⇐⇒ 4 = 4, x < −1
−1 ≤ x ≤ 3 3
x=
−1
− x − (x + 1) = 4 ⇐⇒ 2 − 2x = 4, x = −1
x>3
−3 + x − (x + 1) = 4 ⇐⇒ −4 = 4, x
√ √
x2 + 3x
2
3
= 3,
x2
2
3
= 1.
− (x − 2) + 3x + (x − 2)
≤ −1
u=
√
v = (x2
x2 + 3x
− u
3
− 2) .
v = 3,
u + v = 1,
u=2
√
x2 + 3x = 2
x=1
x=
v=
−1 2
⇐⇒ x
+ 3x = 4,
−4 (x2
x2
v=
− 2 = −1
3
− 2) = −1, x = 1 x = −1 x=1
u=2
−1
× 18 = 222.222.222 12.345.679 × 27 = 333.333.333 12.345.679 × 54 = 666.666.666 12.345.679
999.999.999
12.345.679
250 200
A B
2/3 2/5 3a+6 8
a
x x
2a+10 6
x y xy
15
2
yx x y
z zxz
z
8 3
α1
ax2 + bx + c a, b c
α2
α1 + α2 2 α1 + α2
√ √ √α + √α 4
1
4
2
x2
−3
− 7x − 2c = 0 c
p(x) = 2x4 + bx3 + cx2 + dx + e p(x) = p(1 x)
−
a=0
a b 1.998x2 + ax + b = 0 ax2 + 1.998x + b = 0
x2 + y 2 + z 2 = 3xyz x,y,z
a2 x2 a, b p(n) = 0
2
2
− (b − 2ac)x + c
= 0,
c
n n a, b
(ax
2
− b)
+ (bx
2
− a)
∈Z
= x,
2x2 = x. x2 + 1
[x]
x
∈N
n4 + 4
n>1 n4 + 4 n
n>1
966666555557 966666555558
966666555558 , 966666555559 n
n200 < 5300 . n
10
1 11
2
3
n
11
11
11
· 10 · 10 ··· 10
n
> 100000.
a ax2
− ax + 12 x
x2 1 x2 3x
−
− −
x 3 x+1
x x2 + 9
{}
{}
[x] x + x = 2 x + 10, [x] [5, 83] = 5
[2, 46] = 2 x
x
{x} {x} = x − [x] p
x2 + ax + b = 0 a1 b1 x2 + a1 x + b1 = 0 a2 b2 x2 +a2 x+b2 = 0 x2 + a3 x + b3 = 0 ∆<0
x2 + a1 x + b1 = 0 x2 + a2 x + b2 = 0 a3 b3
x2 = 0
x2 = 0
