## Introduction to Quadratic Equations

Quadratic equations are a core topic in algebra, representing functions of the form:

where , , and are constants, and . These equations are called “quadratic” because the highest power of the variable is two.

Quadratic equations appear in various fields and real-life situations such as physics (projectile motion), finance (calculating profits), and engineering (structural analysis).

## Basic Concept of Factorization

Factorization refers to breaking down a polynomial into simpler terms, or factors, that, when multiplied together, give the original polynomial. For a quadratic equation, this involves expressing it as:

If we can write the quadratic expression in this factored form, solving the equation becomes much more straightforward.

Consider the quadratic equation:

We can factorize it into:

Setting each factor equal to zero gives the solutions:

Hence, the solutions are and .

## Steps to Solve Quadratic Equations by Factorization

Here is a general method to solve quadratic equations by factorization:

**Write the equation in standard form:**Ensure the quadratic equation is in the form .**Find two numbers that multiply to and add to :**These numbers will help in breaking the middle term.**Rewrite the equation:**Break the middle term using the two numbers found.**Factor by grouping:**Group the terms in pairs and factor out the common factors.**Solve for :**Set each factor equal to zero and solve for .

Let’s apply these steps to solve:

1. Write in standard form: It’s already in form.

2. Find two numbers that multiply to and add to . Numbers are 6 and 1.

3. Rewrite equation:

4. Factor by grouping:

5. Solve for :

So, the solutions are and .

## Real-World Application Examples

Quadratic equations often come up in real-world contexts. Here are a few examples:

### Example 1: Projectile Motion

When you throw a ball, its path forms a parabola. The equation of motion can be represented as a quadratic equation:

where is the height, is the time, is the initial velocity, and is the initial height.

To find when the ball hits the ground, set and solve the resulting quadratic equation by factorization.

### Example 2: Area Optimization

Suppose you want to create a rectangular garden with an area of 100 square meters, and you know the length should be 10 meters more than the width. Let be the width, then:

Rewriting it:

Solving this quadratic equation by factorization helps in determining the dimensions.

## Practice Problems

Here are some practice problems for you to test your understanding:

- Solve the quadratic equation by factorization:
- Factorize and solve:
- Find the roots of the equation:

Try to follow the steps and find the solutions. Reviewing these problems with your peers or teacher will help reinforce the concept.

## Conclusion

Solving quadratic equations by factorization is an essential skill in algebra and has diverse applications in real life. By understanding the process and practicing it, you can get proficient in solving these equations. Remember, factorization makes the equation simpler and helps you find the solutions effectively.

Keep practicing, and don’t hesitate to seek help if you encounter difficulties. Mastery of this topic will serve as a strong foundation for more advanced mathematics.