Applications of Quadratic Equations

1. Steps to Solve Quadratic Equation Application Problems
1. Define the unknown : Understand the problem and define the unknown as .
2. Form a quadratic equation : Set up a quadratic equation based on the problem's description.
3. Solve the equation : Solve the quadratic equation for .
4. Verify the solution : Check if the obtained solution satisfies the conditions of the problem.


2. Note:
   When solving quadratic equations, not all solutions may be valid, so you must check if the solution meets the problem's conditions.
   
   
3. Applications Related to Numbers
1. Two consecutive integers: or (where is an integer).
2. Three consecutive integers: (where is an integer).
3. Two consecutive even numbers: or (where is an even number) or (where is a natural number).
4. Two consecutive odd numbers: or (where is an odd number) or (where is a natural number).
5. Three consecutive even (or odd) numbers: or .
6. Sum of natural numbers from 1 to :
   .


4. Applications Related to Objects Thrown Upwards
   When an object is projected upward, its height after seconds is given by . The time when the height is is the solution to the quadratic equation where
   
   
5. Applications Involving Time, Distance, and Speed
   
   
   (Distance) = (Speed) (Time)
   
   
   
   
6. Applications Involving Geometry
1. (Area of a rectangle) (Length) (Width)
2. (Perimeter of a rectangle) Length Width