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The x-intercepts of a graph represent the points where the graph crosses the x-axis. These intercepts are found by determining the values of \(x\) that make \(y = 0\) in the given equation. Essentially, they are the solutions to the equation when the graph meets the x-axis.

To find the x-intercepts of a quadratic equation like \(y = x^2 + x - 2\), you need to set \(y = 0\) and solve the resulting equation: \(0 = x^2 + x - 2\). By solving this, you will find the specific x-values where the graph touches or crosses the axis.

This process results in the potential x-intercepts being the roots of the quadratic equation. In our example, solving \(x^2 + x - 2 = 0\) tells us that the x-values are \(x = 1\) and \(x = -2\). These are the x-intercepts, meaning the graph touches or intersects the x-axis at these points.

The y-intercept of a graph is the point where the graph crosses the y-axis. Unlike x-intercepts, we find the y-intercept by setting \(x = 0\) and solving for \(y\). This process effectively tells us where the graph meets the y-axis by providing the y-coordinate of the intercept.

In the context of quadratic equations, like the example of \(y = x^2 + x - 2\), finding the y-intercept involves a simple calculation. Substitute \(x = 0\) into the equation and solve explicitly for \(y\).

For our specific problem, substituting gives: \(y = 0^2 + 0 - 2 = -2\). Therefore, the graph intersects the y-axis at the point \((0, -2)\). This means the y-intercept is \(-2\), establishing its location on the vertical axis.

Factoring quadratic equations is an algebraic method used to solve equations of the form \(ax^2 + bx + c = 0\). It transforms the quadratic expression into a product of two binomials. This is especially useful for finding solutions to equations, helping to identify x-intercepts.

When factoring, we aim to break down the expression into two simpler expressions such that their product gives back the original quadratic equation. For the equation \(x^2 + x - 2 = 0\), factoring involves finding two numbers whose product is equal to \(-2\) (the constant term) and whose sum is \(1\) (the coefficient of \(x\)).

This results in the factors being \((x - 1)(x + 2)\). These factors give us clues to solve the equation by setting each equal to zero: \(x - 1 = 0\) or \(x + 2 = 0\). Solving these, we find \(x = 1\) and \(x = -2\). Thus, factoring allows us to determine the x-intercepts of the quadratic function efficiently and clearly.