Solving Quadratic Equations: Methods and Examples

Quadratic equations are a fundamental part of algebra and are often encountered in various fields of mathematics, physics, and engineering. The general form of a quadratic equation is (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). This article will explore two different methods to solve quadratic equations: factoring and using the quadratic formula. We will also provide several examples to help illustrate these methods.

Factoring Method

The factoring method involves expressing the quadratic expression as a product of two linear expressions. Let's consider the quadratic equation:

(x^2 - 4x - 4 0)

First, we will use the factoring method to solve this equation.

Example 1: Factoring Method

Given: (x^2 - 4x - 4 0) 1. Rewrite the equation to identify factors of the constant term that add up to the coefficient of the x term. In this case, we need factors of -4 that add up to -4.

We can rewrite the equation as:

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

This can be factored into:

((x - 2)(x - 2) 0)

Factoring out the common term:

((x - 2)(x - 2) 0)

This gives us the solutions:

(x - 2 0) or (x - 2 0)

Hence, the solutions are:

(x 2) or (x 2)

This equation has a double root at (x 2).

Quadratic Formula Method

The quadratic formula is a powerful tool to solve any quadratic equation. The formula is:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

Let's apply this method to the same equation, (x^2 - 4x - 4 0).

Example 2: Using the Quadratic Formula

Given: (x^2 - 4x - 4 0) 1. Identify the values of (a), (b), and (c): (a 1), (b -4), (c -4)

2. Substitute these values into the quadratic formula:

(x frac{-(-4) pm sqrt{(-4)^2 - 4(1)(-4)}}{2(1)} frac{4 pm sqrt{16 16}}{2} frac{4 pm sqrt{32}}{2} frac{4 pm 4sqrt{2}}{2} 2 pm 2sqrt{2})

This gives us two solutions:

(x 2 2sqrt{2}) or (x 2 - 2sqrt{2})

However, in this case, we notice that the discriminant is zero, indicating a repeated root. Hence, the solutions are:

(x 2)

Thus, the quadratic equation has a double root at (x 2).

Conclusion

We have covered two methods to solve quadratic equations: factoring and using the quadratic formula. Both methods are valuable and can be applied depending on the equation at hand. In this example, we encountered a quadratic equation with a double root at (x 2). This emphasizes the importance of choosing the appropriate method based on the specific equation.

Key Takeaways

Factoring method involves expressing the quadratic equation as a product of two linear expressions. The quadratic formula is a universal method applicable to all quadratic equations. A double root occurs when the discriminant is zero.

Keywords: quadratic equations, factoring, quadratic formula, solving equations