if x-y =12 and xy =5 ^1/4 find the value of x+y

Question

Solve it correctly stepwise 

Answer 1

Remember a simple math problem like “The area of a rectangle is 20 and the width is 4. What is the length?” The answer was, very conveniently, the whole number 5. Such nice whole numbers don’t often occur in real life, but they make learning math easier. I call them “magic numbers.”

In terms of polynomials in algebra, this problem has “magic expressions.” Here is some background information that is quite useful:
         (x+y)(x+y) = x² + 2xy + y²              [eq1]
         (x-y)(x-y) = x² - 2xy + y²                 [eq2]

We have givens for (x-y) and for xy. Also, (x+y) can be expressed in terms of (x-y) and xy:

        (x+y)(x+y) = x² - 2xy + y² + 4xy

        (x+y)(x+y) = (x-y)(x-y) + 4xy

        (x+y) = ±√( (x-y)(x-y)+4xy )

Plugging in values:

        (x+y) = ±√( 12*12 + 4*5^(1/4) )

        (x+y) = ± 12.24669

Answer 2

if x-y =12 and xy =5 ^1/4 find the value of x+y

 

First, solve each equation for x:

x = 12 + y

x = 5^1/4 /y

 

Since there are both equal to x, they are equal to each other.

Set them equal and solve for y:

12 + y = 5^1/4 /y     subtract y from both sides

12 = 5^1/4 /y - y      on right, 2nd term needs a common denominator with 1st term

12 = 5^1/4 /y - y² /y     on right, use common denominator to combine numerators

12 = (5^1/4 - y²) /y     multiply both sides by y

12y = 5^1/4 - y²     subtract 5^1/4 from and add y² to both sides

y² + 12y - 5^1/4 = 0    use quadratic formula a=1, b=12, c=-5^1/4

y = [-12 ± √(12² - 4(1)(-5^1/4) ) ] / [2(1) ]     simplify 

y = [-12 ± √144+4(5^1/4) ] / 2     use calculator

y ≈ -12.1233, 0.1233

 

Now find x.

y = -12.1233

x -( -12.1233) = 12

x = -0.1233

 

or

y = 0.1233

x - 0.1233 = 12

x = 12.1233

 

Finally, find x + y.

case 1: x= -0.1233, y= -12.1233

x + y = -0.1233 + -12.1233 = -12.2466

 

case 2: x = 12.1233, y = 0.1233

x + y = 12.1233 + 0.1233 = 12.2466

 

 

check:

case 1:

x-y = -0.1233 -( -12.1233) = 12  and xy = (-0.1233)( -12.1233)= 1.4948 ≈ 5 ^1/4

case 2:

x-y = 12.1233 - 0.1233  = 12 and xy = (12.1233)(0.1233)= 1.4948 ≈ 5 ^1/4