Formation of Quadratic Equations

Formation of Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, often written in the form:

ax2 + bx + c = 0

Where:

  • a, b, and c are coefficients.
  • x represents the variable.

1. General Approach:

  • The formation of quadratic equations often involves translating real-world problems or mathematical relationships into a quadratic equation.

2. Standard Form:

  • Quadratic equations are typically written in standard form, where the variable x is squared, and other terms are either linear or constants.

3. Steps to Form a Quadratic Equation:

a. Identify the Unknown: - Determine what quantity or variable you want to find. This will be represented by x in the equation.

b. Express the Relationship: - Express the given relationship or problem in terms of the unknown quantity x.

c. Set Up the Equation: - Write an equation that represents the given relationship or problem.

d. Rearrange the Equation: - Rearrange the equation so that it is in standard quadratic form: ax2 + bx + c = 0.

e. Assign Coefficients: - Identify and assign values to the coefficients a, b, and c based on the given problem or relationship.

4. Common Scenarios for Formation:

a. Quadratic Patterns: Some problems involve patterns that can be expressed as quadratic equations. For example, the area of a square depends on the length of one side, and this relationship is quadratic.

b. Projectile Motion: Problems involving the motion of objects under the influence of gravity often lead to quadratic equations. The height of an object in free fall is an example.

c. Profit and Revenue: In business and economics, quadratic equations are used to model profit, revenue, and cost functions.

d. Geometry and Area: Problems related to the areas of geometric shapes, such as squares, rectangles, and circles, can lead to quadratic equations.

e. Age Problems: Age-related problems often lead to quadratic equations. These involve the current age, future age, and past age of individuals.

5. Example:

  • Scenario: Suppose you want to find the dimensions of a rectangular garden whose area is 24 square meters. The length is 2 meters more than the width.

  • Step 1: Identify the Unknown: Let's say the width of the garden is x meters.

  • Step 2: Express the Relationship: The length is 2 meters more than the width, so the length is x + 2 meters.

  • Step 3: Set Up the Equation: The area of the rectangle is given by the product of its length and width, so:

    Area = Length × Width

           24 = (x + 2) × x

  • Step 4: Rearrange the Equation: To bring it to standard quadratic form, expand the right side of the equation:

    24 = x2+ 2x

  • Step 5: Assign Coefficients: Here, a = 1 (the coefficient of x2), b = 2 (the coefficient of x), and c = -24 (the constant term).

  • So, the quadratic equation is: x2 + 2x - 24 = 0.

6. Formation of Quadratic Equations from Given Roots

Steps to Form Quadratic Equations from Given Roots:

  1. Identify the Roots: The first step is to identify and understand the given roots. Roots are the values of x that satisfy the quadratic equation. If you're given the roots α and β, you have:

    • α: One of the solutions.
    • β: The other solution.
  2. Use the Sum and Product of Roots:

    • For a quadratic equation ax2 + bx + c = 0, the sum of the roots is -b/a, and the product of the roots is c/a. These relationships are derived from Vieta's formulas.
  3. Write the Equation:

    a. To form the quadratic equation when the roots are given, start with the basic structure:

    a(x - α)(x - β) = 0

    b. Expand this equation by multiplying the binomials:

    a(x2 - (α + β)x + αβ) = 0

  4. Determine the Values of a, b, and c:

    • You can match the coefficients with those in the original equation. For instance:
      • a matches with the coefficient of x2 in the original equation.
      • -b/a matches with the coefficient of x.
      • αβ matches with the constant term c in the original equation.
  5. Write the Final Quadratic Equation:

    • Substitute the values of a, b, and c that you've determined into the general form ax2 + bx + c = 0 to get the specific quadratic equation based on the given roots.

Example: Suppose you are given the roots α = 3 and β = -2 and asked to form the quadratic equation.

Step 1: Identify the roots: α = 3, β = -2.

Step 2: Use the sum and product of roots:

  • The sum of roots: α + β = 3 - 2 = 1
  • The product of roots: αβ = 3 * (-2) = -6

Step 3: Write the equation:

  • a(x - α)(x - β) = 0
  • a(x2 - (α + β)x + αβ) = 0

Step 4: Determine the values of a, b, and c:

  • a matches with the coefficient of x2 (1 in this case).
  • -b/a matches with the coefficient of x (0 in this case).
  • αβ matches with the constant term (c) (-6 in this case).

Step 5: Write the final quadratic equation:

  • a(x2 - x - 6) = 0
  • Substituting the values:
  • (x2 - x - 6) = 0 is the quadratic equation based on the given roots.

7. Solve the Quadratic Equation:

  • You can solve the quadratic equation using methods such as factoring, the quadratic formula, or completing the square to find the values of x that satisfy the given problem or relationship.