Question

Use any method to solve the system of nonlinear equations. Please show steps. Thank you.

x² + y² = 1
y = 20x² - 1

Answer 1

Here given that x^2 + y^2 =1 --------(1)

And y=20x^2 -1 ----------(2)

Using the value of y from (2) in (1) we have

x^2 + (20x^2 -1)^2 =1

Or, x^2 + 400x^4 -40x^2 +1=1

Or, 400x^4 -39x^2 =0

Or, x^2 ( 400x^2 -39)=0

Or, x=0,0 and 400x^2-39=0

Or, x=√(39)/20 , -√(39)/20

So when x=0, from (1) we have y=1

And when x=√(39)/20 from (1) we have

39/400 + y^2 =1

Or, y^2 = 361/400

Or, y=19/20, -19/20

So the solutions are x=0,y=1; x=√(39)/20, y=19/20; x=-√(39)/30,y=19/20; x=√(39)/20,y=-19/20; x=-√(39)/20,y=-19/20