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Understanding how to solve quadratic equations is essential in algebra. These equations are of the form \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants.

To solve these equations, there are multiple methods, including factoring, using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), or completing the square. Each of these methods provides a way to find the x-values (roots) where the parabola represented by the quadratic equation intersects the x-axis.

When working with piecewise functions like the textbook problem, it's important to remember that the solution for the quadratic part depends on the interval of 'x' we are dealing with. Factoring is often the simplest approach when possible, but if the equation is not easily factorable, completing the square or the quadratic formula becomes very handy.

Quadratic functions can be expressed in the vertex form, which is \( y = a(x-h)^2 + k \), where '(h, k)' is the vertex of the parabola, and 'a' determines the direction and width of the parabola.

This form is particularly useful because it clearly shows the maximum or minimum point (the vertex) of the parabola. In the context of the given piecewise function, the vertex form allows us to easily identify the turning points of each segment of the graph. If 'a' is positive, the parabola opens upwards and the vertex represents the lowest point. If 'a' is negative, as in the second piece of the piecewise function, the parabola opens downwards and the vertex is the highest point.

Completing the square is a method used to solve quadratic equations and to convert them into vertex form. It involves adding and subtracting a particular value to create a perfect square trinomial within the equation.

To complete the square for the quadratic equation \( ax^2 + bx + c \), one must first divide all terms by 'a' (if 'a' is not equal to 1), resulting in \( x^2 + \frac{b}{a}x + \frac{c}{a} \). Then add and subtract \( (\frac{b}{2a})^2 \), and organize the terms to get \( (x + \frac{b}{2a})^2 \) on one side, moving the constant to the other side. This process reveals the vertex \( -\frac{b}{2a}, c - (\frac{b}{2a})^2 \) and converts the equation into the vertex form.

This method is particularly useful for graphing parabolas, as it quickly identifies the vertex of the parabola, making it easier to sketch the graph accurately.

Graphing parabolas is a critical skill when dealing with quadratic functions. A parabola is the graph of a quadratic function and has a characteristic 'U' shape, either opening upward or downward based on the coefficient of \( x^2 \).

For a piecewise function that includes quadratic functions, each quadratic section will form a segment of a parabola. To graph these, first determine the direction the parabola opens by the sign of 'a'. If 'a' is positive, it opens upwards. If 'a' is negative, it opens downwards. Next, locate the vertex using either the vertex form or by completing the square. Finally, plot additional points by choosing x-values and solving for 'y', then draw the shape of the parabola through these points.

In the provided exercise, the first quadratic opens upwards with the vertex at (0,5), while the second opens downwards. It's important to apply the piecewise constraints when graphing, using only the portion of the parabola that falls within the given x-value ranges.