How do you solve the system #4x^2-56x+9y^2+160=0# and #4x^2+y^2-64=0#?

2 Answers
Jul 9, 2016

#y^2 = 64 - 4x^2#

#y =+- sqrt(64 - 4x^2)#

#4x^2 - 56x + 9(sqrt(64 - 4x^2))^2 + 160 = 0#

#4x^2 - 56x + 9(64 - 4x^2) + 160 = 0#

#4x^2 - 56x + 576 - 36x^2 + 160 = 0#

#0 = 32x^2 + 56x - 736#

#0 = 8(4x^2 + 7x - 92)#

#0 = 8(4x^2 - 16x + 23x - 92)#

#0 = 8(4x(x - 4) + 23(x - 4))#

#0 = 8(4x + 23)(x - 4)#

#x = -23/4 and 4#

Case 1:

#4(-23/4)^2 + y^2 - 64 = 0#

#4(529/16) + y^2 - 64 = 0#

#y^2 = 64 - 529/4#

#y = O/#

Case 2:

#4(4)^2+ y^2 - 64 = 0#

#4(16) + y^2 - 64 = 0#

#y^2 = 64 - 64#

#y^2 = 0#

#y = 0#

The only real solution is #x = 4#, #y = 0#.

Thus, the solution set is #{-4, 0}#.

Hopefully this helps!

Jul 9, 2016

#y = 0# and #x = 4#

Explanation:

Making #x_2=x^2,y_2=y^2# and solving for #x_2,y_2# the system

#{ (4 x_2+9y_2=56x-160), (4x_2+y_2=64) :}#

we obtain

# { (x_2 = 1/4 (92 - 7 x)), (y_2 = 7 x-28) :}#

Now we solve

#x^2 = 1/4 (92 - 7 x)#

obtaining #x = {-23/4,4}#

and

#y = pm sqrt(7x-28)#

or

#y = pm sqrt(7 cdot 4-28) = 0# excluding complex solutions.

we have

#y = 0# and #x = 4#

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