📐 Step-by-Step Derivation of the Quadratic Formula 🔗

If you’ve studied algebra, you have likely memorized the quadratic formula. But where does it come from?

The quadratic formula isn’t just magic; it is the result of solving the standard quadratic equation for $x$ using a technique called completing the square.

Here is the step-by-step derivation, explaining exactly how we get from the standard form to the famous formula.

📑 Table of Contents 🔗

1️⃣ Step 1: Start with the Standard Form 🔗

Every quadratic equation can be written in the standard form, where $a$, $b$, and $c$ are known constants, and $a \neq 0$:

2️⃣ Step 2: Divide by a 🔗

To use the “completing the square” method easily, we want the coefficient of our $x^2$ term to be $1$. We achieve this by dividing the entire equation (both sides) by $a$:

3️⃣ Step 3: Move the Constant Term 🔗

Next, we isolate the terms with $x$ on the left side. We do this by subtracting the constant term, $c/a$, from both sides of the equation:

4️⃣ Step 4: Complete the Square 🔗

This is the most crucial step. We want to turn the left side of the equation into a “perfect square trinomial”—an expression that can be factored into the shape $(x + k)^2$.

To do this, we take exactly half of the $x$ coefficient (which is $b/a$), square it, and add it to both sides to keep the equation balanced.

Half of $b/a$ is $b/(2a)$. Squaring that gives us $\left(\frac{b}{2a}\right)^2$. Let’s add it to both sides:

5️⃣ Step 5: Factor the Left Side & Square the Right Side 🔗

Because of how we constructed it in Step 4, the left side can now be factored perfectly into squared binomial form. On the right side, we will simply apply the square to our fraction:

6️⃣ Step 6: Find a Common Denominator 🔗

To combine the fractions on the right side of the equation, we need a common denominator. The lowest common denominator for $a$ and $4a^2$ is $4a^2$.

We multiply the numerator and denominator of $-c/a$ by $4a$:

Now we can combine them into a single fraction. We’ll also rearrange the numerator to look a bit more familiar:

7️⃣ Step 7: Take the Square Root 🔗

To solve for $x$, we need to get rid of the square on the left side. We do this by taking the square root of both sides.

Crucial rule: When you take the square root of both sides of an equation, you must account for both the positive and negative roots ($\pm$) on the right side.

We can simplify the denominator since the square root of $4a^2$ is simply $2a$:

8️⃣ Step 8: Isolate x 🔗

The final step is to get $x$ completely by itself. We subtract $b/(2a)$ from both sides:

Because both fractions already share the same denominator ($2a$), we can combine them into one final, unified expression:

🎯 Conclusion 🔗

And there it is! By systematically completing the square, we have derived the Quadratic Formula. This formula will now allow you to solve for $x$ in any quadratic equation simply by plugging in the values of $a$, $b$, and $c$.

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