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Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They are called quadratic because "quad" means square, and the highest power of the variable \( x \) in these equations is 2. Quadratic equations are quite common in mathematics and have a variety of applications.
To solve these equations, you often need to rearrange and simplify the equation, just as we did in the original exercise by moving terms around. The goal is to isolate \( x^2 \) on one side. Once you achieve this, taking the square root of both sides will help you find the variables' values.
One thing to remember is that when you take the square root, you need to consider both the positive and negative roots. This comes from the fact that both \((-a)(-a)\) and \((a)(a)\) will give \(a^2\). Therefore, when dealing with quadratic equations, your solutions may often have two values.

• Quadratic equations usually have two solutions.
• Solutions are found by isolating \(x\) and taking the square root.
• Always consider both \( + \) and \( - \) square roots.

In a rational function, such as the one we considered, \( f(x) = \frac{a(x)}{b(x)} \), the expression is made up of a numerator and a denominator. The numerator is the top part of the fraction, and the denominator is the bottom part.
For functions to be defined, the denominator cannot be zero because division by zero is undefined in mathematics. To find the zeros of a rational function, you set the numerator equal to zero and solve for \( x \). This is because when the numerator is zero, the entire fraction is zero, provided the denominator is not zero.
Knowing the role and restrictions of numerators and denominators helps in solving rational functions efficiently:

• The numerator must be zero for the function to have zeros.
• The denominator must not be zero, otherwise, the function is undefined.

Solving equations involves finding the values of the variable that make the equation true. In the context of finding zeros of a rational function, solving equations means simplifying the function to isolate the variable \( x \).
The key steps include:

• Set the relevant part of the function (numerator for zeros) equal to zero.
• Simplify and rearrange terms, ensuring you maintain the balance of the equation.
• Use algebraic methods like factoring or extracting square roots to isolate the variable.

These tactics help you find what values of \( x \) satisfy the equation. Knowledge of solving equations translates across many areas of math, such as algebra and calculus, and is a foundational skill for any math student.
Understanding the process will make solving even complex equations a lot more manageable. Simplifying equations is about breaking them down into easy, manageable steps and staying organized throughout the process.