Solving Quadratic Equations: A Step-by-Step Guide with Practical Examples

Mathematics is often seen as a challenging subject, especially when it comes to solving quadratic equations. However, by breaking down the process into manageable steps, you can effectively tackle even the most complex equations.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree. It is generally written in the form:

ax2 bx c 0

Where a, b, and c are constants, and a ≠ 0. In this example, we will solve the equation x2 - 2x - 48 0.

Step-by-Step Solution

Let's go through the process of solving x2 - 2x - 48 0 using a popular method:

Step 1: Expand and Rearrange

Since there's no need for expansion in this case, we directly move all the terms to one side:

x2 - 2x - 48 0

Step 2: Factor the Quadratic Equation

Our goal is to find two numbers that multiply to -48 (the constant term) and add to -2 (the coefficient of x). After some trial and error, we find that: 8 and -6.

We can then factorize the equation as:

(x - 8)(x 6) 0

Step 3: Set Each Factor Equal to Zero

To find the values of x, we set each factor to zero:

x - 8 0 or x 6 0

Step 4: Solve for x

From x - 8 0, we get:

x 8

From x 6 0, we get:

x -6

Alternative Methods

There are several alternative methods to solve quadratic equations, which may also be useful in different contexts:

Completing the Square Method

Besides factoring, you can also use the completing the square method:

x2 - 2x 48

Add 1 to both sides to complete the square:

x2 - 2x 1 49

This simplifies to:

(x - 1)2 49

Taking the square root of both sides, we get:

x - 1 7, x - 1 -7

Thus, the solutions are:

x 8, x -6

Intuitive Approach

Another straightforward approach is to place the equation in its quadratic form: