Wavy Curve Method for Quadratic Inequalities

Wavy Curve Method for Finding Roots of Polynomials

The Wavy Curve Method, also known as the Method of Intervals, is a strategy used to solve inequalities of the form $$f(x) \geq 0$$ or $$f(x) \leq 0$$. This method is particularly useful for solving quadratic inequalities, which are the same as solving quadratic equations. The technique involves the following steps:

1. Find the roots of the polynomial: Identify the roots of the given polynomial equation, denoted as $$x = \alpha_1, x = \alpha_2, x = \alpha_3, \dots, x = \alpha_n$$.

2. Plot the roots on the number line: Arrange the roots of the polynomial equation in increasing order and plot them on the number line.

3. Draw a wavy curve: Starting from the right of the last root, draw a wavy curve along the number line that alternatively changes direction at the roots of the polynomial equation. The direction of the curve depends on the parity (odd or even) of the power of the corresponding root.

4. Identify the intervals: The Wavy Curve Method is a positive function for all the intervals in which the curve lies above the x-axis.

Here's an example to illustrate the Wavy Curve Method:

Consider the quadratic inequality $$\frac{x(3 - x)}{(x + 4)^2} \geq 0$$.

1. Factor the polynomial: $$\frac{x(3 - x)}{(x + 4)^2} = \frac{x(3 - x)}{(x + 4)^2}$$.

2. Find the roots of the polynomial: In this case, the roots are $$x = 0$$ and $$x = 3$$.

3. Plot the roots on the number line: Place the roots on the number line, starting from the right of the last root ($$x = 3$$) and moving left towards the first root ($$x = 0$$).

4. Draw a wavy curve: Starting from the right of $$x = 3$$, draw a wavy curve along the number line that changes direction at $$x = 0$$ and $$x = 3$$, depending on the parity of the power of each root.

5. Identify the intervals: The Wavy Curve Method is a positive function for all the intervals in which the curve lies above the x-axis. In this case, the curve lies above the x-axis for $$x \in [0, 3]$$, so the solution set is $$x \in [0, 3]$$.

In summary, the Wavy Curve Method is a powerful technique for solving inequalities involving polynomials, particularly quadratic inequalities. By finding the roots of the polynomial, plotting them on the number line, and drawing a wavy curve that changes direction at the roots, we can identify the intervals where the function is positive or negative and solve the inequality.