## Profile

Matematyka - Matematyczny Świat wokół nas.
1,398 followers|3,849,428 views

## Stream

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

Find all possible solutions (real numbers) for (x, y) = ?﻿
·  Translate
1

·  Translate

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

Find all possible solutions (real numbers) for (x, y) = ?﻿

Find all possible solutions (real numbers) for (x, y) = ?﻿

2

{5∛(x+y) = 6 + 2√(x-y)
{18 - 3√(x-y) = 6∛(x+y)
{∛(x+y) = u
{√(x-y) = v
{5u = 6 + 2v
{18 – 3v = 6u
{5u - 2v = 6  /·6
{-6u – 3v =-18  / · 5
{30u - 12v = 36
{-30u – 15v =-90
30u – 30u -12v -15v = 36 – 90
-27v = -54
v = 2
{5u - 2v = 6  /·3
{-6u – 3v =-18  / · (-2)
{15u - 6v = 18
{12u +6v =36
15u + 12u -6v +6v = 18 +36
27u = 54
u= 2
{u= 2
{v=2
{∛(x+y) = u
{√(x-y) = v
{∛(x+y) = 2    /³
{√(x-y) = 2    /²
{x+y = 8
{x-y = 4
x+x+y-y=8+4
2x = 12
x = 6

{x+y = 8 /·(-1)
{x-y = 4
{-x-y = -8
{x-y = 4
-x+x-y-y=-8+4
-2y = -4
y = 2
Solution:
{x = 6
{y=2﻿

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

Find all possible solutions (real numbers) for (x, y) = ?﻿
·  Translate
1
1

·  Translate

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

Find all possible solutions (real numbers) for (x, y) = ?

Find all possible solutions (real numbers) for (x, y) = ?

7
1

oh crap. sorry  ﻿

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.﻿
1
·  Translate

### Matematyka - Matematyczny Świat wokół nas.owner

Pytania matematyczne (Math Questions)  -

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.

http://www.matematyczny-swiat.pl/2014/06/srednica-okregu-wpisanego-w-trapez.html﻿

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.

http://www.matematyczny-swiat.pl/2014/06/srednica-okregu-wpisanego-w-trapez.html﻿
1

|AB| + |CD| = |AD| + |BC|
a + b = |AD| + |AD| => |AD| = |BC|
a + b = 2|AD|
(a +b)/2 = |AD|

|AE| = (|AB| - |CD|)/2
|AE| = (a –b)/2

|ED|²  = |AD|² - |AE|²
|ED|² = [(a +b)/2]² - [(a –b)/2]²
|ED|² = (a + b)²/4  + (a – b)²/4
|ED|² = [(a + b)² - (a – b)²]/4
|ED|² = [a² + 2ab + b² -( a² - 2ab + b²)]/4
|ED|² = [a² + 2ab + b² - a² + 2ab - b²]/4
|ED|² = 4ab/4
|ED|² = ab
|ED| = √ab

|ED| = |FG| = |KL| = √ab﻿
·  Translate
In their circles
4,376 people
Have them in circles
1,398 people

### Matematyka - Matematyczny Świat wokół nas.owner

Pytania matematyczne (Math Questions)  -

Find all possible solutions (real numbers) for (x, y) = ?﻿

Find all possible solutions (real numbers) for (x, y) = ?﻿

1

{5∛(x+y) = 6 + 2√(x-y)
{18 - 3√(x-y) = 6∛(x+y)
{∛(x+y) = u
{√(x-y) = v
{5u = 6 + 2v
{18 – 3v = 6u
{5u - 2v = 6  /·6
{-6u – 3v =-18  / · 5
{30u - 12v = 36
{-30u – 15v =-90
30u – 30u -12v -15v = 36 – 90
-27v = -54
v = 2
{5u - 2v = 6  /·3
{-6u – 3v =-18  / · (-2)
{15u - 6v = 18
{12u +6v =36
15u + 12u -6v +6v = 18 +36
27u = 54
u= 2
{u= 2
{v=2
{∛(x+y) = u
{√(x-y) = v
{∛(x+y) = 2    /³
{√(x-y) = 2    /²
{x+y = 8
{x-y = 4
x+x+y-y=8+4
2x = 12
x = 6

{x+y = 8 /·(-1)
{x-y = 4
{-x-y = -8
{x-y = 4
-x+x-y-y=-8+4
-2y = -4
y = 2
Solution:
{x = 6
{y=2﻿

### Matematyka - Matematyczny Świat wokół nas.

Math Questions  -

Find all possible solutions (real numbers) for (x, y) = ?﻿

5
5

x=6;y=2﻿

### Matematyka - Matematyczny Świat wokół nas.owner

Pytania matematyczne (Math Questions)  -

Find all possible solutions (real numbers) for (x, y) = ?

Find all possible solutions (real numbers) for (x, y) = ?

1

{ y^(x² + 7x + 12) = 1
{ x + y = 6
=>
y^(x² + 7x + 12) = 1
y^(x² + 7x + 12) = y^0
x² + 7x + 12 = 0 <=>
∆ = b² - 4ac
∆ = 7² - 4 · 1 · 12 => a=1, b=7, c=12
∆ = 49 -48
∆ = 1
√∆ = 1
x₁ = (-b-√∆)/2a = (-7-1)/2 = -8/2 = -4
x₁  + y₁ = 6 => y₁ = 6 - x₁ = 6 –(-4) = 10
{x₁ = -4
{y₁ = 10
x₂ = (-b+√∆)/2a = (-7+1)/2 = -6/2 = -3
x₂ + y₂ = 6 => y₂ = 6 - x₂ = 6 –(-3) = 9
{x₂ = -3
{y₂ = 9

y^(x² + 7x + 12) = 1 <=> y = 1
y = 1 =>  x + y = 6
x = 6 – y
x = 6 -1
x = 5
{x₃ = 5
{y₃ = 1
y^(x² + 7x + 12) = 1
y=1 =>
L = y^(x² + 7x + 12)
L = 1^(5² + 7·5 + 12)
L = 1^(25 + 35 + 12)
L = 1^72
L = 1,   R=1, L=R

y^(x² + 7x + 12) = 1 <=> y = -1
y = -1 =>  x + y = 6
x = 6 – y
x = 6 –(-1)
x = 7
{x₄= 7
{y₄ = -1
y^(x² + 7x + 12) = 1
y=-1 =>
L = y^(x² + 7x + 12)
L = (-1)^(7² + 7·7 + 12)
L = (-1)^(49 + 49 + 12)
L = (-1)^110
L = 1,   R=1, L=R

Solutions:
(-4, 10), (-3, 9), (5, 1), (7, -1)
﻿

### Matematyka - Matematyczny Świat wokół nas.

Math Questions  -

Find all possible solutions (real numbers) for (x, y) = ?

30
19

Easy..﻿

### Matematyka - Matematyczny Świat wokół nas.

Shared publicly  -

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.

http://www.matematyczny-swiat.pl/2014/06/srednica-okregu-wpisanego-w-trapez.html﻿

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.

http://www.matematyczny-swiat.pl/2014/06/srednica-okregu-wpisanego-w-trapez.html﻿
4

Ok, thanks.﻿

### Matematyka - Matematyczny Świat wokół nas.

Math Questions  -

We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.

http://www.matematyczny-swiat.pl/2014/06/srednica-okregu-wpisanego-w-trapez.html﻿
9
6

|AB| + |CD| = |AD| + |BC|
a + b = |AD| + |AD| => |AD| = |BC|
a + b = 2|AD|
(a +b)/2 = |AD|

|AE| = (|AB| - |CD|)/2
|AE| = (a –b)/2

|ED|²  = |AD|² - |AE|²
|ED|² = [(a +b)/2]² - [(a –b)/2]²
|ED|² = (a + b)²/4  + (a – b)²/4
|ED|² = [(a + b)² - (a – b)²]/4
|ED|² = [a² + 2ab + b² -( a² - 2ab + b²)]/4
|ED|² = [a² + 2ab + b² - a² + 2ab - b²]/4
|ED|² = 4ab/4
|ED|² = ab
|ED| = √ab

|ED| = |FG| = |KL| = √ab﻿
·  Translate
People
In their circles
4,376 people
Have them in circles
1,398 people
Communities
Created by Matematyka - Matematyczny Świat wokół nas.
View all
Story
Tagline
Blog matematyczny Minor! Matematyka - Matematyczny Świat wokół nas.
Introduction
Autor: Robert Karolewski

"Powiedz mi, a zapomnę. Pokaż mi, a zapamiętam. Pozwól mi zrobić, a zrozumiem. (Konfucjusz)"

Matematyka jest skarbcem bankiera.
Piękno matematyki jest w tym, że wszystko można opisać matematycznie!