91Ó°ÊÓ

Linear equations are mathematical statements where each term is either a constant or the product of a constant and a single variable. In essence, they are algebraic expressions that form a straight line when graphed on a Cartesian plane. Consider the standard form of a linear equation: \(ax + b = 0\). Here, \(a\) and \(b\) are constants, and \(x\) is the variable we're solving for. A linear equation essentially shows the relationship between two variables.

• Characteristics: The graph is always a straight line.
• The equation has a constant rate of change, known as the slope.
• It can have either one solution, infinite solutions, or no solution.

In the context of our exercise, we do not deal directly with linear equations but rather with determinants of a matrix that lead us to a quadratic equation, which we solve to find specific values for \(t\). Hence, linear principles are foundational in understanding the relationships and transformations needed to solve these types of problems.

Quadratic equations are expressions that can be generally written in the form \(ax^2 + bx + c = 0\). They are called quadratic because the term "quad" generally means square, indicating the square of the variable \(x\). Here, \(a\), \(b\), and \(c\) are constants.

• Characteristics: The highest power of the variable is 2.
• These equations typically have two solutions, which can be found by factorization, completing the square, or using the quadratic formula.
• The graph of a quadratic equation is a parabola.

In solving the exercise given, we use the product of solving a quadratic equation by factorization. For example, expanding the matrix determinant leads us to equations like \((t - 2)(t - 11) = 0\), which upon solving gives us the values \(t = 2\) and \(t = 11\). Each solution represents a point where the determinant of the matrix equals zero, indicating a change in sign or a critical point in the function.

Factorization refers to breaking down a complex expression into simpler terms or factors that when multiplied give back the original expression. In the context of quadratic equations, factorization involves rewriting the equation in a product form, where each factor could be set to zero to find the values of the variable.

• Method: Look for two numbers that multiply to the constant term \(c\) and add up to \(b\).
• Rewrite the equation using these numbers to break down the middle term.
• Factor by grouping and solve for the variable.

In our exercise, factorization plays a critical role in finding the values of \(t\) that make the determinant zero. For example, the equation \(t^2 - 13t + 22 = 0\) becomes \((t - 2)(t - 11) = 0\) after factorization. This approach simplifies solving the quadratic equation by transforming it into a product of linear terms, making it easier to solve for the variable \(t\).