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A quadratic expression typically takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, with \(a \eq 0\). It's called 'quadratic' because 'quadra' refers to the square, and the defining feature of these expressions is the \(x^2\) term.
Understanding how to manipulate and solve quadratic expressions is crucial in mathematics. They can reveal points where the expression equals zero, known as 'roots' or 'solutions'.
To solve a quadratic expression like the one in our exercise \( (x-3)^2 > 0 \), you may be tempted to expand it; however, it's more efficient to recognize this as a 'perfect square'. Perfect squares are expressions where the variable portion is squared and the expression will always yield non-negative values for any real number input, except for its root.
Quadratic expressions can be graphed as parabola-shaped curves on a coordinate plane. For our example, \( (x-3)^2 \) represents a parabola opening upwards with its vertex at \( (3,0) \) and it does not cross the x-axis—it touches it exactly at one point, the root. Thus, the parabola is always above the x-axis except for this single point, reflecting that the expression is always positive save for when \( x = 3 \).

Interval notation is a compact way of writing inequalities and solutions involving ranges of values. It's commonly used to communicate where an expression is positive, negative, or takes on specific values.
In interval notation, we use brackets and parentheses to represent closed and open intervals, respectively. For instance, \[a, b\] means all numbers from \(a\) to \(b\), inclusive. On the other hand, \(a, b\) indicates all the numbers between \(a\) and \(b\), but not including \(a\) or \(b\).

Using Infinity in Interval Notation

Infinity (\(\infty\)) and negative infinity (\(-\infty\)) are used to describe boundless intervals. For example, the solution to the inequality \( (x-3)^2 > 0 \) means \(x\) can be any number except for 3. The interval notation for this is \( (-\infty, 3) \cup (3, \infty) \), where the union symbol (\(\cup\)) means 'or' and indicates that the solution can be in either interval. Since we cannot reach infinity and \(x\) cannot equal 3, the parentheses indicate that these endpoints are not included in the intervals.
Interval notation is essential for clearly presenting the range of solutions to inequalities and is especially useful when dealing with continuous sets of numbers.

When dealing with a quadratic inequality like \( (x-3)^2 > 0 \), the solutions we find are not just single values but rather entire ranges of values that satisfy the inequality. Solving inequalities involves finding these ranges and expressing them, often through interval notation.

Steps to Solve Inequalities

• Find the critical points by setting the quadratic expression equal to zero and solving for \(x\).
• Test intervals around these critical points to determine where the inequality holds true.
• Express the solution using interval notation to capture all the \(x\) values that make the inequality valid.

With our example, the process revealed that the inequality is true for all real numbers except where \(x = 3\). Therefore, considering all numbers less than 3 and greater than 3 gives us the complete solution. The critical step is to ensure that the intervals chosen do indeed make the original inequality a true statement.
In essence, the process of solving inequalities can be thought of as a search for where a graph of the expression lies above or below the x-axis when we're looking at 'greater than' (>) or 'less than' (<) inequalities, respectively. This graphical viewpoint can be particularly helpful when trying to visualize the solution set for an inequality.