Mathematics · Algebra

Deriving the
Quadratic Formula

A complete, step-by-step proof — from the general equation to the formula you already know.

📐 6 Steps · 🔑 Completing the Square · ⏱ 5 min read

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We want to solve the general quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$. The goal is to isolate $x$ using only algebraic manipulation — no guessing, no shortcuts.

Step 01

Divide Everything by $a$

To set up for completing the square, we need the coefficient of $x^2$ to be exactly $1$. Dividing the entire equation by $a$ achieves this:

$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$

Step 02

Move the Constant to the Right Side

Isolate the terms containing $x$ on the left so we have room to complete the square:

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

Step 03

Complete the Square

We want the left side to look like a perfect square $(x + \text{something})^2$. The rule is: take half the coefficient of $x$, then square it:

$$\text{coefficient of } x = \frac{b}{a} \implies \frac{b}{2a} \implies \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$$

Add this value to both sides of the equation:

$$x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$$

The left side is now a perfect square trinomial:

$$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$

Why does this work? Because $(x + k)^2 = x^2 + 2kx + k^2$. Matching terms, $2k = \tfrac{b}{a}$, so $k = \tfrac{b}{2a}$ and $k^2 = \tfrac{b^2}{4a^2}$.

Step 04

Simplify the Right Side

Combine the two fractions on the right by giving them a common denominator of $4a^2$:

$$-\frac{c}{a} = -\frac{4ac}{4a^2}$$

So the equation becomes:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$

Step 05

Take the Square Root of Both Sides

Apply the square root to both sides. Remember: taking the square root introduces a $\pm$ because both the positive and negative roots are valid solutions.

$$x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$$

Since $4a^2$ is a perfect square, $\sqrt{4a^2} = 2|a|$. Because the $\pm$ already handles both signs, we write $2a$ by convention:

$$x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$$

Step 06

Isolate $x$

Subtract $\dfrac{b}{2a}$ from both sides:

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

Both fractions already share the denominator $2a$, so we can combine them into one:

$$\boxed{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$$

📋 The Full Derivation at a Glance

$$ax^2 + bx + c = 0$$

÷ a↓

$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$

move constant↓

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

complete the square↓

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$

square root both sides↓

$$x + \frac{b}{2a} = \pm\frac{\sqrt{b^2-4ac}}{2a}$$

isolate x↓

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

💡

Key Takeaway: The quadratic formula is simply completing the square applied to the general case. You don't need to memorize the derivation — but understanding it means you will never forget the formula, because you can always re-derive it yourself from scratch.