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# System

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How many solutions of the system x - y = 3, x^2 - y = 3 are there such that both x and y are real?

Jul 5, 2022

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Convert the second equation into a quadratic: $$y = x^2 - 3$$

Substitute it into the first equation: $$-x^2 + x + 3 = 3$$

Simplify: $$-x^2 + x = 0$$

Note that because the discriminant ($$1 - (4 \times 1 \times 0) = 1$$) is positive, there will be exactly $$\color{brown}\boxed{2}$$ real solutions.

Jul 5, 2022