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If x + y =2 and xy = 23, then what is x^2 + y^2?

 Jan 29, 2015

Best Answer 

 #3
avatar+94548 
+5

Using x + y = 2   →  y = 2 - x      and xy = 23  ...so....

x(2- x)= 23      expand

2x - x^2 = 23    rearrange

x^2 - 2x + 23 = 0    .....which implies that...

x^2 = 2x -23

And y^2 = (2 - x)^2 = (x -2)^2 = x^2 - 4x + 4

Therefore

x^2 + y^2 =

2x - 23 + x^2 - 4x + 4 =

x^2 - 2x - 19

 

 Jan 29, 2015
 #1
avatar+17747 
+5

Square both sides of the equation  x + y  = 2:

--->   (x + y)²  =  2²

Multiplying out:

--->  x² + 2xy + y²  =  4

Since xy = 23:

--->   x² + 23 + y²  =  4

--->   x² + y²  =  -19            (This does not have real answers!)

 Jan 29, 2015
 #2
avatar+20848 
+5

If  x + y =2  and  xy = 23 ,  then what is  x^2 + y^2  ?

$$(x+y)^2=x^2+2xy+y^2 \quad | \quad -2xy\\\\
x^2+y^2 =(\underbrace{x+y}_{=2})^2-2\underbrace{xy}_{=23}\\\\
x^2+y^2 = 2^2-2*23\\\\
x^2+y^2 = 4-46\\\\
x^2+y^2 = -42$$

.
 Jan 29, 2015
 #3
avatar+94548 
+5
Best Answer

Using x + y = 2   →  y = 2 - x      and xy = 23  ...so....

x(2- x)= 23      expand

2x - x^2 = 23    rearrange

x^2 - 2x + 23 = 0    .....which implies that...

x^2 = 2x -23

And y^2 = (2 - x)^2 = (x -2)^2 = x^2 - 4x + 4

Therefore

x^2 + y^2 =

2x - 23 + x^2 - 4x + 4 =

x^2 - 2x - 19

 

CPhill Jan 29, 2015

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