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Find all possible solutions (real numbers) for (x, y) = ?
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{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
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Find all possible solutions (real numbers) for (x, y) = ?
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Claudio Ambrosio Madeira's profile photoFaizan Khan's profile photoRobert Karolewski's profile photoAlireza Ghaffari's profile photo
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Kabir Agrawal's profile photoRobert Karolewski's profile photoVictor Arjona's profile photoAdriana Safaniuk's profile photo
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oh crap. sorry +Victor Arjona 
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We have a data length of the bases of an isosceles trapezium. Calculate the diameter KL of a circle.
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Matematyka - Matematyczny Świat wokół nas.
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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
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Mazin Kalil's profile photoRobert Karolewski's profile photo
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|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
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Matematyka - Matematyczny Świat wokół nas.
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Pytania matematyczne (Math Questions)  - 
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Robert Karolewski's profile photo
 
{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
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Edgar Cortez Machorro's profile photoJoel Tan's profile photoRobert Karolewski's profile photoadrian santos's profile photo
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x=6;y=2
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Matematyka - Matematyczny Świat wokół nas.
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Pytania matematyczne (Math Questions)  - 
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Robert Karolewski's profile photoZofia Kornaga's profile photo
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{ 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)
 
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Find all possible solutions (real numbers) for (x, y) = ?

http://www.matematyczny-swiat.pl/2014/06/ukad-rownan.html
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Poerwohadi poer's profile photoNazrah Mulaffer's profile photoPasha Balakhonov's profile photoAyush Vikram's profile photo
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Easy..
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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
Carlos Warley's profile photoRobert Karolewski's profile photo
3 comments
 
Ok, thanks.
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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
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Mazin Kalil's profile photomarija hrubik's profile photoEugeny Kornienko's profile photoRobert Karolewski's profile photo
4 comments
 
|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
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