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To understand function evaluation, consider it as plugging a specific value into a function and determining the result. In the original exercise, the function is defined as \( h(x) = x^2 - 4x - c \). Here, \( x \) acts as a placeholder that can be replaced by any number. When we perform function evaluation, we substitute this placeholder with a specific value to see what the function outputs.

For example, if you replace \( x \) with \( c \) in \( h(x) \), the expression becomes \( h(c) = c^2 - 4c - c \), which simplifies to \( c^2 - 5c \). This evaluation is crucial for identifying special values of parameters, like finding a specific \( c \) such that \( h(c) = c \). The goal is to simplify and solve the expression for the missing variable, which in this case is \( c \).

Understanding function evaluation allows you to transform and solve equations by correctly substituting and simplifying variables. This method is extensively used in calculus and algebra to explore how functions behave and interact.

Factoring is a powerful method used to simplify quadratic equations into products of simpler terms, making it easier to solve. A quadratic equation usually has the form \( ax^2 + bx + c = 0 \). In our exercise, after substitution and simplification, we reached the equation \( c^2 - 6c = 0 \).

The goal of factoring is to express the quadratic as a product of two binomials. For the equation \( c^2 - 6c = 0 \), you can factor by pulling out the common term. Here, \( c(c - 6) = 0 \). This factorization breaks down the problem into simpler equations \( c = 0 \) or \( c - 6 = 0 \).

Factoring utilizes properties of numbers and algebra to break complex expressions into manageable parts, making it straightforward to identify solutions. It is important to remember that not all quadratics factor neatly, and sometimes other methods like completing the square or using the quadratic formula are necessary. However, when possible, factoring provides a quick and efficient path to solving quadratic equations.

Solving equations involves finding the values of variables that make the equation true. Once a quadratic equation is factored, it can be solved easily by setting each factor equal to zero. In our exercise, we had the equation \( c(c-6) = 0 \). From this, we had two potential solutions: \( c = 0 \) or \( c - 6 = 0 \).

To solve \( c - 6 = 0 \), simply add 6 to both sides, resulting in \( c = 6 \). Solving these equations told us the points where the original function intersects the x-axis, if viewed on a graph.

Yet, as the condition specified a nonzero \( c \), the only valid solution is \( c = 6 \). Solving equations accurately demands careful attention to arithmetic operations and a clear understanding of mathematical properties. This approach allows us to isolate variables and find the values that satisfy given conditions or constraints.