For the following system's solutions, which of the choices represents the correct y-value(s) used in the solution? Hint: make a sketch of the system first so that you know how many solutions to expect.x2 + y2 = 4y = x2 - 1Select one:−12+13√2−12+132−12−13√2−12−132−12+13√2and−12−13√2−12+132𝑎𝑛𝑑−12−13212+13√212+132
Question
For the following system's solutions, which of the choices represents the correct y-value(s) used in the solution? Hint: make a sketch of the system first so that you know how many solutions to expect.x2 + y2 = 4y = x2 - 1Select one:−12+13√2−12+132−12−13√2−12−132−12+13√2and−12−13√2−12+132𝑎𝑛𝑑−12−13212+13√212+132
Solution
The question seems to be asking for the y-values of the solutions to the system of equations:
x² + y² = 4 y = x² - 1
To solve this system of equations, we can substitute the second equation into the first to get:
x² + (x² - 1)² = 4 This simplifies to: x² + x⁴ - 2x² + 1 = 4 Rearranging terms gives us: x⁴ - x² - 3 = 0
This is a quadratic equation in terms of x². We can solve it by factoring, completing the square, or using the quadratic formula. The solutions to this equation will give us the x-values of the solutions to the system of equations.
Once we have the x-values, we can substitute them into the second equation to get the corresponding y-values.
However, without the options to choose from, it's impossible to say which of the given choices represents the correct y-values. Please provide the options so I can give a more accurate answer.
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