Determine the number of real solution(s) for:
$10x + y = -x^2 + 2$
[ ] real solution(s)
The equation given is \(10x + y = -x^2 + 2\). Our goal is to determine the number of real pairs \((x, y)\) that satisfy this equation. We begin by rearranging the equation to solve for \(y\) in terms of \(x\). By subtracting \(10x\) from both sides, we obtain the expression:
\[y = -x^2 - 10x + 2\]
This equation defines a function \(y = f(x) = -x^2 - 10x + 2\), which represents a parabola in the Cartesian plane.
To determine the number of real solutions, we look at the nature of the function \(f(x) = -x^2 - 10x + 2\). For any real number \(x\), the expression \(-x^2 - 10x + 2\) evaluates to a unique real number \(y\). Since the domain of the function is all real numbers (\(x \in \mathbb{R}\)), there exists a corresponding \(y\) value for every possible input \(x\). Because there are infinitely many real numbers \(x\), there must be infinitely many corresponding real pairs \((x, y)\) that satisfy the equation.
Since the equation defines a functional relationship where \(y\) is determined uniquely for every real number \(x\), and there is no constraint that limits the domain of \(x\), we conclude that the number of real solutions \((x, y)\) is infinite. The final answer is \(\boxed{\text{infinitely many}}\).
In coordinate geometry, curves like parabolas are sets of infinitely many points. For instance, if you were mapping the trajectory of a ball thrown into the air, the path could be modeled by a quadratic equation like this. At any given horizontal position \(x\), there is a specific vertical position \(y\), forming a continuous curve consisting of an infinite number of points that represent all possible positions of the ball throughout its flight.