Deriving the Quadratic Formula: A Step-by-Step Guide

In mathematics, one of the most fundamental and widely used equations is the quadratic formula, which provides the solutions to quadratic equations in standard form. Understanding how this formula is derived is not only crucial for solving specific equations but also deepens our understanding of algebraic techniques. Let's explore the derivation of the quadratic formula from a quadratic equation in standard form.

Quadratic Equation in Standard Form

A quadratic equation can be written in the standard form:

ax2 bx c 0, where a, b, and c are constants (a ≠ 0).

Steps to Derive the Quadratic Formula

Step 1: Dividing by a

If a ≠ 1, the first step is to divide every term in the equation by a:

x2 frac{b}{a}x frac{c}{a} 0

Step 2: Rearranging the Equation

Rearrange the equation to isolate the quadratic term on one side:

x2 frac{b}{a}x -frac{c}{a}

Step 3: Completing the Square

To complete the square, the goal is to transform the left side of the equation into a perfect square trinomial. Here are the steps to achieve this:

Take half of the coefficient of x, which is frac{b}{a}, and square it: (frac{b}{2a})2 frac{b^2}{4a^2}.

Add frac{b^2}{4a^2} to both sides of the equation to maintain its balance:

x2 frac{b}{a}x frac{b^2}{4a^2} -frac{c}{a} frac{b^2}{4a^2}

Factor the left side of the equation into a perfect square trinomial:

left(x frac{b}{2a}right)2 -frac{c}{a} frac{b^2}{4a^2}

Step 4: Simplifying the Right Side

To simplify the right side of the equation, find a common denominator for frac{c}{a} and frac{b^2}{4a^2}:

left(x frac{b}{2a}right)2 frac{b^2 - 4ac}{4a^2}

Step 5: Taking the Square Root of Both Sides

Take the square root of both sides to solve for x, introducing the plus-or-minus sign:

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

Step 6: Isolating x

Simplify the right side of the equation:

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

Isolate x by subtracting frac{b}{2a} from both sides:

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

Step 7: Combining Terms Over a Common Denominator

Combine the terms over a common denominator to simplify the expression:

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

Conclusion

The derived quadratic formula: