The Quadratic Formula: Derivation and Usage

The Quadratic Formula is a fundamental tool in algebra that provides a straightforward method for finding the roots of a quadratic equation. This article explains the derivation of the Quadratic Formula using the method of completing the square, along with a comprehensive overview of its significance in solving quadratic equations.

Introduction to Quadratic Equations and the Quadratic Formula

A quadratic equation is an equation of the form ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. The Quadratic Formula, x frac{-b ± sqrt{b^2 - 4ac}}{2a}, provides the solutions to any quadratic equation, and is derived through a process known as completing the square.

Deriving the Quadratic Formula

Let's start with the standard form of a quadratic equation: ax^2 bx c 0. Our goal is to manipulate this equation so that we can solve for x.

Step 1: Divide Both Sides by a

First, we divide every term in the equation by a to simplify the equation: x^2 frac{b}{a}x frac{c}{a} 0. We can rewrite this as: x^2 frac{b}{a}x -frac{c}{a}.

Step 2: Move frac{c}{a} to the Right Side

Next, we move the constant term frac{c}{a} to the right side of the equation: x^2 frac{b}{a}x -frac{c}{a}.

Step 3: Complete the Square

To complete the square, we need to add a specific value to both sides of the equation. This value is left(frac{b}{2a}right)^2, which is half of the coefficient of x squared. Adding this value to both sides, we get:

x^2 frac{b}{a}x left(frac{b}{2a}right)^2 left(frac{b}{2a}right)^2 - frac{c}{a}

Now, the left-hand side is a perfect square trinomial: left(x frac{b}{2a}right)^2.

Step 4: Take the Square Root of Both Sides

With the left side as a perfect square, we can take the square root of both sides:

sqrt{left(x frac{b}{2a}right)^2} pm sqrt{left(frac{b}{2a}right)^2 - frac{c}{a}}

This simplifies to:

x frac{b}{2a} pm sqrt{frac{b^2}{4a^2} - frac{c}{a}}

Combining the terms under a common denominator under the square root:

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

Step 5: Solve for x

Finally, we solve for x by subtracting frac{b}{2a} from both sides:

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

This results in the Quadratic Formula:

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

A Practical Application: Solving Quadratics

The Quadratic Formula is not just a theoretical tool; it is widely used in practical applications. For instance, it is essential in physics for solving problems related to motion and trajectories, in engineering for design optimization, and in economics for cost and revenue analysis.

Conclusion

In summary, the Quadratic Formula is a powerful tool that originates from the method of completing the square. It simplifies the process of solving quadratic equations, making it a valuable part of the algebraic toolkit. Understanding and knowing how to derive this formula isn't just academic; it has practical implications in various fields.