91Ó°ÊÓ

Understanding how to solve quadratic equations is crucial in algebra. These are equations of the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). There are various methods to solve these equations: factoring, using the quadratic formula, or completing the square.

For instance, given a function \(h(x)=\frac{10}{x^{2}-2x}\), the denominator \(x^{2}-2x\) can be set to zero to find the undefined points of the function. This results in a quadratic equation to solve. By factoring or any other method, we can find that \(x=0\) and \(x=2\) are the roots of the equation, meaning these values would make the function undefined. When approaching quadratic equations, it's important to first look for the most straightforward method, such as factoring, before moving on to more complex approaches.

Factorization is the process of breaking down an expression into a product of simpler factors that, when multiplied together, give the original expression. This process is particularly useful when solving quadratic equations, as it can transform complex expressions into more manageable ones.

For example, consider the quadratic equation from the previous section, \(x^{2}-2x=0\). We can factor out an \(x\), resulting in \(x(x-2)=0\). This reveals that \(x=0\) and \(x=2\) would make the original function from the exercise undefined. Essentially, factorization turns the problem of finding where a function is undefined into a simpler problem of finding zeros of its factors.

A function becomes undefined when its output is not a real number or cannot be determined. This often happens due to operations like division by zero, which is not permissible in mathematics. In our case, the function \(h(x)=\frac{10}{x^{2}-2x}\) becomes undefined at certain \(x\) values.

Identifying these values involves setting the denominator equal to zero and solving the resulting equation. As we determined, when \(x=0\) or \(x=2\), the denominator becomes zero, causing the function to be undefined. Recognizing when functions are undefined is a key skill when studying their properties, such as their domain.

Interval notation is a way of describing a range of numbers along the number line. It is concise and useful for expressing domains or ranges of functions. In interval notation, the use of parentheses \( ( ) \) indicates that an endpoint is not included, while brackets \( [ ] \) indicate inclusivity.

The solution to our previous problem expressed in interval notation would omit the values of \(x\) that make the function undefined, resulting in \( (-\infty, 0) \cup (0, 2) \cup (2, \infty) \). This notation tells us that the function is defined for all real numbers except for \(x=0\) and \(x=2\), which are the points of discontinuity where the function is undefined.