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Understanding speed and distance calculations is foundational in solving problems related to moving objects. The relationship between speed, distance, and time can be understood using the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] This formula implies that if you know any two of the three variables—distance, speed, and time—you can calculate the third one. In our problem, we used it to find the time taken for the turbo-jet and the super-prop plane to cover their respective distances. Remember:

• Distance covered is directly proportional to speed when time is constant.
• For a constant distance, time varies inversely with speed.

In practical terms, this means that faster speeds result in shorter travel times for the same distance.

Quadratic equations appear frequently in algebra, especially in problems involving squared terms. A quadratic equation typically has the form: \[ ax^2 + bx + c = 0 \] In our exercise, we derived a quadratic equation from the relationship between the times taken by the planes: \[ 3s^2 - 650s + 140000 = 0 \] To solve it, we used the quadratic formula: \[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, a is 3, b is -650, and c is 140000. By calculating the discriminant \[ b^2 - 4ac \], we found the solutions for s. Key points:

• The discriminant (b^2 - 4ac) determines the nature of the roots.
• If it’s positive, we get two real roots; if zero, one real root; if negative, the roots are complex.

For our equation, the positive root gave us the speed of the super-prop plane.

Solving algebraic word problems requires translating the given information into mathematical expressions. Here are steps to approach such problems:
1. **Read the problem carefully**: Understand what is being asked. In our case, we need to find the speeds of two planes.
2. **Define the variables**: Decide what each variable represents. We let s be the speed of the super-prop plane, making the turbo-jet's speed s + 50.
3. **Set up equations**: Translate the conditions in the problem into algebraic equations. We used the given distances and times to form our initial equations.
4. **Solve the equations**: Use algebraic methods to solve the equations. We solved a quadratic equation using the quadratic formula.
5. **Interpret the solution**: Make sure your solution makes sense in the context of the problem. We confirmed our speeds were reasonable for planes.
These steps help break down complex problems into manageable parts, making it easier to find a solution.