91Ó°ÊÓ

Quadratic equations are a type of polynomial equation in which the highest exponent of the variable is 2. The general form of a quadratic equation is \[ ax^2 + bx + c = 0 \] where \( a, b, \) and \( c \) are constants and \( x \) is the variable. Solving quadratic equations often involves using the quadratic formula: \[ x = \frac{-b \,\pm \,\sqrt{b^2 - 4ac}}{2a} \] In this problem, we ended up with the quadratic equation \[ t^2 - 3t - 2 = 0 \] representing the relationship between the time taken by two workers when combined versus individually. We solved it using the quadratic formula, which gave us the value of \( t \). This helps determine how long the faster worker will take alone.

Work rate problems involve determining the amount of work done over time by different entities working together or individually. In our problem, each worker has a specific rate at which they can complete a task. To solve these types of problems, we use the formula: \[ \text{Work rate} = \frac{1}{\text{Time}} \] For example, if the slower worker takes \( t + 1 \) hours, their work rate is \( \frac{1}{t + 1} \). When combining work rates, you add individual rates together:\[ \frac{1}{t} + \frac{1}{t+1} \] Their combined rate must equal the total work done (one lawn) over the combined time (2 hours). Hence, we get: \[ \frac{1}{t} + \frac{1}{t+1} = \frac{1}{2} \]

Math problem-solving involves several steps to break down and approach a problem logically. Here's a simple strategy that we applied in this exercise:

• Define the variables: Assign symbols to unknowns to simplify the representation.
• Translate words to equations: Convert descriptions into mathematical expressions.
• Combine equations: Use relevant formulas to merge and simplify them.
• Solve for the unknown: Use algebraic techniques such as the quadratic formula.
• Verify your solution: Substitute back to ensure correctness.

By following these steps methodically, we can tackle most algebra word problems and work rate problems efficiently. The key is to stay organized and work through each part systematically.