Introduction to Quadratic Equations

Introduction to Quadratic Equations

Quadratic equations are a crucial part of algebra, forming the foundation for solving complex problems across various disciplines. Let's dive into what quadratic equations are, explore their methods of solution, and review real-life applications.

What is a Quadratic Equation?

A quadratic equation is any equation that can be expressed as:

Where:

• are constants ().
• is the variable.

Standard Form of Quadratic Equations

The standard form of a quadratic equation is:

Key components:

• : The quadratic term.
• : The linear term.
• : The constant term.

Example:

Convert into standard form:
Rearrange terms: .
Multiply by : .

Methods to Solve Quadratic Equations

1. Factorization

This involves expressing the quadratic equation as a product of two binomials.

Example: Solve .
Steps:

1. Factorize: .
2. Solve: , .

Answer: .

2. Completing the Square

This technique transforms the quadratic equation into a perfect square trinomial.

Example: Solve .
Steps:

1. Rewrite: .
2. Add : .
3. Factorize: .
4. Solve: .
   • , .

Answer: .

3. Quadratic Formula

The quadratic formula is:

Example: Solve .
Steps:

1. Identify .
2. Discriminant: .
3. Apply formula:
   
   .

Solutions: .

4. Graphical Method

In this method, the equation is plotted as a parabola. The roots are the points where the parabola intersects the x-axis.

Example: For :

1. Plot .
2. Intersection points: .

Answer: .

Nature of Roots

The discriminant () determines the type of roots:

• : Two distinct real roots.
• : One repeated root.
• : Complex roots.

Example: Solve .

1. Discriminant: .
2. Complex roots:
   
   .

Answer: .

Applications of Quadratic Equations

1. Projectile Motion: Calculating the path of an object in free fall.
2. Geometry: Finding dimensions or areas of geometric shapes.
3. Optimization: Maximizing profits or minimizing costs in economics.

Example:
A rectangle's area is 24 m². If the length is 2 m more than the width, find the dimensions.

1. Let width = . Length = .
2. Area: .
3. Solve : .
4. (positive value).

Answer: Width = 4 m, Length = 6 m.

Sample Questions with Answers

1. Solve .
   Answer: .
2. Find the nature of roots for .
   Answer: , one repeated root ().
3. Solve using the quadratic formula: .
   Answer: .

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