Solving Quadratic Equations Using Factorization: A Step-by-Step Guide

In this article, we will explore two methods to solve the quadratic equation (x^2 - 4x - 3 0). The first method involves using the factorization technique, while the second uses the quadratic formula. We will also discuss the benefits and applications of each method, ensuring that you understand the intricacies of solving quadratic equations.

Introduction to Quadratic Equations

A quadratic equation is an equation of the second degree in one variable. It can be written in the standard form as (ax^2 bx c 0), where (a), (b), and (c) are constants. The unique feature of a quadratic equation is that it can have up to two solutions.

Solving by Factorization Method

The factorization method is particularly useful when the quadratic equation can be easily expressed as a product of two binomials. Let's solve the equation (x^2 - 4x - 3 0) using this method.

Step-by-Step Guide to Factorization

Identify the quadratic equation: The given equation is (x^2 - 4x - 3 0). Here, (a 1), (b -4), and (c -3). Factor the quadratic: We need to find two numbers that multiply to (c -3) and add up to (b -4). The numbers -3 and -1 satisfy these conditions because: -3 * -1 3 -3 (-1) -4 Express the quadratic as a product of two binomials:

(x^2 - 4x - 3 (x - 3)(x 1) 0)

Set each factor to zero:

(x - 3 0 Rightarrow x 3)

(x 1 0 Rightarrow x -1)

Final solution: The solutions to the equation (x^2 - 4x - 3 0) are (x 3) and (x -1).

Solving by Quadratic Formula Method

If the factorization method is not straightforward, the quadratic formula is a reliable alternative. The quadratic formula is given by: [x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Applying the Quadratic Formula

Quadratic formula application: Using the equation (x^2 - 4x - 3 0), we have (a 1), (b -4), and (c -3).

Calculate the discriminant:

[D b^2 - 4ac (-4)^2 - 4(1)(-3) 16 12 28]

Find the solutions:

[x frac{-(-4) pm sqrt{28}}{2(1)} frac{4 pm sqrt{28}}{2} frac{4 pm 2sqrt{7}}{2} 2 pm sqrt{7}]

Final solution: The solutions are (x 2 sqrt{7}) and (x 2 - sqrt{7}).

Comparison and Discussion

Using the factorization method, we obtained the solutions (x 3) and (x -1). However, the quadratic formula provided the solutions (x 2 sqrt{7}) and (x 2 - sqrt{7}). It is important to note that both methods are valid, but the factorization method is generally faster and more straightforward when the equation can be factored easily. Additionally, factorization gives solutions within the real domain, which is often desirable for practical applications.

Conclusion

In this article, we have explored the factorization method and the quadratic formula for solving the quadratic equation (x^2 - 4x - 3 0). The factorization method, while not always applicable, is often the preferred method due to its simplicity and the real domain solutions it provides. The quadratic formula, on the other hand, offers a general approach that can handle a wider range of quadratic equations, including those with complex solutions.

Keywords

Quadratic equations Factorization method Solving quadratic equations