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The substitution method is a powerful tool for solving systems of equations. It involves solving one of the equations for one variable, then substituting that expression into the other equation. This allows you to reduce the number of variables. In our example, we began with two equations involving variables \(x\) and \(y\):

• \(x^2 + (y-2)^2 = 4\)
• \(x^2 - 2y = 0\)

To use substitution, we first solve the second equation for \(x^2\), which gives \(x^2 = 2y\). This is then substituted into the first equation, replacing \(x^2\), to form a new equation in terms of \(y\) only. This method effectively reduces the problem to a simpler one-variable equation, making it easier to solve.

Quadratic equations are polynomial equations of degree two, typically in the form \(ax^2 + bx + c = 0\). These equations can have two solutions due to their parabolic graph shape. In the exercise, substituting \(x^2 = 2y\) into \((x^2 + (y-2)^2 = 4)\), simplifies the problem to a quadratic equation in \(y\):

• \(3y^2 - 4y = 0\)

This is a standard form of a quadratic equation, easily recognizable and solvable by various methods such as factoring, completing the square, or applying the quadratic formula. Here, we choose to factor the equation, because it is one of the most straightforward methods and often requires simple algebraic manipulation.

Factoring quadratics is one of the simplest and most intuitive methods to solve quadratic equations. It involves breaking down a quadratic equation into a product of two binomials. The key step in the example was rearranging \((3y^2 - 4y = 0)\) so that it can be expressed as a product:

• \(y(3y - 4) = 0\)

By setting each factor equal to zero, \(y = 0\) and \(3y - 4 = 0\), we can easily determine the solutions for \(y\). This method works particularly well for equations that can be factored into integers or simple expressions, as it quickly reveals the roots or solutions of the equation.

The concept of intersection of curves refers to finding the common points where two curves meet on a graph. In the context of solving a system of equations, each equation represents a curve. The solutions to the system are the points at which these curves intersect. In this exercise, one equation represents a circle and the other a parabola. These distinct curves intersect at:

• \((x=0, y=0)\)
• \((x=\sqrt{8/3}, y=4/3)\)
• \((x=-\sqrt{8/3}, y=4/3)\)

These points signify the solutions to the system, indicating where the algebraic expressions satisfy both equations simultaneously. Understanding this concept helps visualize how two or more equations relate in their graphical form and how solutions can be comprehended as tangible points of intersection in space.