Deriving the Quadratic Formula Through Completing the Square

Understanding and deriving the quadratic formula is a fundamental part of algebra. This article walks you through the step-by-step process of obtaining the famous quadratic formula by completing the square on the generic quadratic equation (ax^2 bx c 0). By following these detailed instructions, you'll gain a deeper understanding of this essential concept.

Introduction to the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically written as (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The standard form of a quadratic equation sets everything to zero, making it easier to manipulate and solve. This article will guide you through the process of deriving the quadratic formula step-by-step.

Step-by-Step Derivation of the Quadratic Formula

Step 1: Start with the Standard Form of a General Quadratic Equation

The standard form of a quadratic equation is: ax^2 bx c 0

Here, (a), (b), and (c) are coefficients that can be any real numbers, and (a eq 0). It's crucial to work with the general form to find a solution applicable to any values of (a), (b), and (c).

Step 2: Subtract (c) from Both Sides

Isolate the (x^2) and (x) terms by subtracting (c) from both sides, resulting in: ax^2 bx -c

Step 3: Divide Both Sides by (a)

This step aims to make the coefficient of (x^2) equal to 1. After dividing both sides by (a), we get: x^2 frac{b}{a}x -frac{c}{a}

Step 4: Complete the Square

To complete the square, add and subtract the same value on the left side to create a perfect square trinomial. The value to add is (left(frac{b}{2a}right)^2). Thus, we have: begin{align*}x^2 frac{b}{a}x left(frac{b}{2a}right)^2 left(frac{b}{2a}right)^2 - frac{c}{a} left(x frac{b}{2a}right)^2 frac{b^2}{4a^2} - frac{c}{a}end{align*}

Next, write the right side under a common denominator: begin{align*}left(x frac{b}{2a}right)^2 frac{b^2 - 4ac}{4a^2}end{align*}

Step 5: Take the Square Root of Both Sides

When taking the square root of both sides, remember that both the positive and negative roots must be considered. Thus, we get: begin{align*}x frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a}end{align*}

Finally, isolate (x) by subtracting (frac{b}{2a}) from both sides: begin{align*}x -frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a}end{align*}

The final equation is: begin{align*}x frac{-b pm sqrt{b^2 - 4ac}}{2a}end{align*}

This is the quadratic formula, which can be used to solve any quadratic equation in the standard form.

Steps Explained in Detail

Step 1: Start with the Standard Form

Any equation with an (x^2) term qualifies as a quadratic equation. The standard form sets everything to zero, enabling easier manipulation. For example, the equation (ax^2 bx c 0) can accommodate any values of (a), (b), and (c), with (a eq 0).

Step 2: Subtract (c) from Both Sides

The goal is to isolate the (x) terms. Subtracting (c) from both sides gives us:

This step prepares the equation for the next manipulation.

Step 3: Divide Both Sides by (a)

Dividing both sides by (a) to ensure the coefficient of (x^2) is 1: begin{align*}x^2 frac{b}{a}x -frac{c}{a}end{align*}

This step simplifies the equation and makes it easier to complete the square.

Step 4: Complete the Square

To complete the square, add and subtract (left(frac{b}{2a}right)^2) on the left side. This creates a perfect square trinomial: begin{align*}x^2 frac{b}{a}x left(frac{b}{2a}right)^2 left(frac{b}{2a}right)^2 - frac{c}{a} left(x frac{b}{2a}right)^2 frac{b^2}{4a^2} - frac{c}{a}end{align*}

Finally, express the right side under a common denominator: begin{align*}left(x frac{b}{2a}right)^2 frac{b^2 - 4ac}{4a^2}end{align*}

Step 5: Take the Square Root of Both Sides

When taking the square root of both sides, include both the positive and negative roots: begin{align*}x frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a}end{align*}

This results in: begin{align*}x -frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a}end{align*}

Step 6: Isolate (x)

Subtract (frac{b}{2a}) from both sides to isolate (x): begin{align*}x frac{-b pm sqrt{b^2 - 4ac}}{2a}end{align*}

This is the quadratic formula, which allows you to solve any quadratic equation.

Conclusion

By following these steps, you can derive the quadratic formula, offering a systematic approach to solving quadratic equations. Understanding the process of completing the square is not only useful in mathematics but also in various fields, including physics, engineering, and economics.