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The distributive property is a fundamental principle in algebra that allows us to break down expressions and make them easier to handle. It's all about multiplying a single term across the terms inside a parenthesis. In the exercise, we have expressions like \( \frac{1}{2}c(2-c) \) and \( \frac{2}{5}c(c+1) \). To simplify these,

• multiply \( \frac{1}{2}c \) with each term inside the parenthesis \((2-c)\), resulting in \( \frac{1}{2}c \times 2 - \frac{1}{2}c \times c \).
• For \( \frac{2}{5}c(c+1) \), multiply \( \frac{2}{5}c \) with \(c\) and \(1\).

By following the distributive property, we simplify the expressions to make the equations more manageable. This principle is handy not just in solving equations but in many areas of algebra, ensuring we maintain the balance and equality of equations.

Simplifying expressions involves breaking down complex algebraic expressions into a simpler, more digestible form. This process is crucial for reaching an easily understandable equation. In our exercise, after using the distributive property, we're left with terms like \( c - \frac{1}{2}c^2 \) and \( 2c^2 + 2c \). The next steps involve:

• Combining like terms: Look for terms with the same variables and exponents. Combine them to simplify the equation.
• Reorganizing the equation: Ensure all terms are on one side if needed, often setting up for solving the equation.

Simplifying helps in reducing errors and in seeing the equation more clearly, which can make subsequent steps like solving for the variable considerably easier. The goal is always to create a straightforward path to finding the solution.

The quadratic formula is a helpful tool for finding the solutions to quadratic equations, which generally have the form \( ax^2 + bx + c = 0 \). When an equation is simplified to this standard form, like \( \frac{5}{2}c^2 + c - \frac{1}{10} = 0 \), the next step is to identify \( a \), \( b \), and \( c \). In this problem, they are \( \frac{5}{2} \), \( 1 \), and \( -\frac{1}{10} \) respectively. The quadratic formula states: \[c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here’s how you can use it:

• Plug in the values of \( a \), \( b \), and \( c \) into the formula.
• Calculate the discriminant (\( b^2 - 4ac \)) to determine the nature of the roots.
• Follow through the operations to solve for \( c \).

This formula gives us the values of \( c \) for which the equation holds true. It's a reliable method ensuring that you get the correct roots even when other methods like factoring are difficult. In this instance, it results in two possible values for \( c \), highlighting the quadratic equation's nature of having two potential solutions.