SOLUTION: y=x^2-10 9x^2-y^2=90 The solution set is?

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Question 985922: y=x^2-10
9x^2-y^2=90
The solution set is?
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(53765) About Me  (Show Source):
You can put this solution on YOUR website!
.
system%28y=x%5E2-10%2C%0D%0A9x%5E2-y%5E2=90%29.

Substitute  y = x%5E2 - 10  from the first equation into the second equation.  You will get

9%2Ax%5E2 - %28x%5E2-10%29%5E2 = 90.

Solve it:

9x%5E2 - (x%5E4+-+20%2Ax%5E2+%2B+100) = 90,

9x%5E2 - x%5E4 + 20x%5E2 - 100 = 90,

x%5E4 - 29x%5E2 + 190 = 0.

It is a bi-quadratic equation.     Don't be afraid !     Don't panic !

The Math gives you an opportunity to demonstrate your skills ! :)

Introduce new variable  z = x%5E2.

Then you will get quadratic equation

z%5E2 - 29z + 190 = 0.

Its roots are  (use the quadratic formula)

z%5B1%5D = 19, z%5B2%5D = 10.

Hence,  x = +/-sqrt%2819%29, +/-sqrt%2810%29.

Then from the first equation of the system

y = x%5E2 - 10 = 19 - 10 = 9   OR   y = 10 - 10 = 0.

Answer.  The solutions are the pairs  (x,y) = (sqrt%2819%29, 9),  (-sqrt%2819%29, 9),  (sqrt%2810%29, 0)  and  (-sqrt%2810%29, 0).


Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
y=x%5E2-10
9x%5E2-y%5E2=90
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y=x%5E2-10
9x%5E2-90=y%5E2
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y=x%5E2-10
y=sqrt%289x%5E2-90%29
----------------
y=x%5E2-10
y=sqrt%289%28x%5E2-10%29%29
----------------
y=x%5E2-10
y=3sqrt%28x%5E2-10%29
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if left sides same, then we have
x%5E2-10=3sqrt%28x%5E2-10%29
x%5E2%2F3-10%2F3=sqrt%28x%5E2-10%29............square both sides
%28x%5E2%2F3-10%2F3%29%5E2=%28sqrt%28x%5E2-10%29%29%5E2
%28x%5E2%2F3%29%5E2-2%28x%5E2%2F3%29%2A%28-10%2F3%29%2B%28-10%2F3%29%5E2=x%5E2-10
%28x%5E4%2F9%29-%2820x%5E2%2F9%29%2B%28100%2F9%29=x%5E2-10.....both sides multiply by 9
9%28x%5E4%2F9%29-9%2820x%5E2%2F9%29%2B9%28100%2F9%29=9x%5E2-90
x%5E4-20x%5E2%2B100=9x%5E2-90
x%5E4-20x%5E2%2B100-9x%5E2%2B90=0
x%5E4-29x%5E2%2B190=0....factor
x%5E4-19x%5E2-10x%5E2%2B190=0
%28x%5E4-19x%5E2%29-%2810x%5E2-190%29=0
x%5E2%28x%5E2-19%29-10%28x%5E2-19%29=0
%28x%5E2-19%29%28x%5E2-10%29+=+0
zeros:

if x%5E2-19+=+0=>x%5E2=19+=>x=sqrt%2819%29+ or x=-sqrt%2819%29+
if x%5E2-10+=+0=>x%5E2=10+=>x=sqrt%2810%29+ or x=-sqrt%2810%29+
approximately:
x=4.4+ or x=-4.4+
x=3.2+ or x=-3.2+
y=x%5E2-10
9x%5E2-y%5E2=90
for x+=+sqrt%2819%29=> y=%28sqrt%2819%29%29%5E2-10=> y=19-10=>y+=+9
for x+=+sqrt%2810%29=> y=%28sqrt%2810%29%29%5E2-10=> y=10-10=>y+=+0
for x+=+sqrt%2819%29=>9%28sqrt%2819%29%29%5E2-90=y%5E2=>171-90=y%5E2=>81=y%5E2=>y+=+9 and =>y+=+-9
for x+=+sqrt%2810%29=>9%28sqrt%2810%29%29%5E2-90=y%5E2=>90-90=y%5E2=>9=y%5E2=>y+=+0
check: