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Quadratic equations are fundamental elements in algebra that help us solve various problems. They are equations where the highest exponent of the variable is 2, often written in the standard form as \(ax^2 + bx + c = 0\). Understanding this form is crucial for solving these equations efficiently. It reveals the structure consisting of three key components: \(ax^2\), \(bx\), and \(c\).

Here's a simple breakdown of each term:

• \(ax^2\) - This is the quadratic term. It defines the parabolic shape of the graph.
• \(bx\) - The linear term affects the slope and horizontal shift of the parabola.
• \(c\) - The constant term moves the parabola vertically up or down the y-axis.

To solve a quadratic equation, you typically bring all terms to one side, resulting in zero on the other side. This is crucial for further solving techniques like factoring. Converting our given equation \(x(2x - 13) = -6\) involves rearranging it to the standard quadratic form, resulting in \(2x^2 - 13x + 6 = 0\). This adjusted form paves the way for subsequent solving steps, such as factoring.

Factoring is one of the most straightforward methods for solving quadratic equations. It involves expressing the quadratic equation as a product of two binomial expressions. This method is particularly effective when the quadratic equation is factorable into whole numbers.

Let's consider the equation transformed into its standard form: \(2x^2 - 13x + 6 = 0\). The next goal is to factor it by splitting the middle term. The search is for two numbers that multiply to give the product of the coefficient of \(x^2\) term and the constant term, which is \((2 \times 6) = 12\), and these same numbers should add up to the coefficient of the \(x\) term, which is \(-13\).

The numbers \(-12\) and \(-1\) satisfy these conditions:

• \((-12) \times (-1) = 12\)
• \((-12) + (-1) = -13\)

This allows us to rewrite:

• \(2x^2 - 12x - 1x + 6 = 0\)

Then factor by grouping:

• \((2x^2 - 12x) + (-1x + 6) = 0\)
• \(2x(x - 6) - 1(x - 6) = 0\)

The common factor \((x - 6)\) is pulled out, leading us to \((2x - 1)(x - 6) = 0\).

Finally, by setting each factor to zero, you find the solutions for \(x\):

• \(x = \frac{1}{2}\)
• \(x = 6\)

Polynomial equations, of which quadratic equations are a special case, involve powers of variables. These equations take forms such as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_0 = 0\). In these equations, \(n\) represents the highest degree or power of the variable, indicating the number of potential solutions or roots the equation can have.

A quadratic equation is specifically a second-degree polynomial equation. Solving polynomial equations generally requires techniques such as:

• Factoring when possible, which is the method used in our example.
• Using the quadratic formula for more complex equations.
• Graphical methods to visualize and approximate solutions.

Understanding how to transform and solve quadratic equations aids in tackling broader polynomial equations. Solving these equations serves as a fundamental skill because it leads to understanding polynomial functions, their graphs, and various applications in science and engineering. By mastering the factoring approach, a foundation is built for advances into more complex polynomial equations and concepts.