Investigation 6: Deriving the Quadratic Formula

New Concept

The most common way of solving a quadratic equation is to use the quadratic formula

What’s next

Next, you’ll use the technique of completing the square to derive this powerful formula step-by-step, revealing how it was constructed.

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To find the roots of any quadratic equation in the form $ax^2 + bx + c = 0$, you can use the quadratic formula:

To solve $2x^2 + 6x - 3 = 0$, identify $a=2, b=6, c=-3$.
$x = \frac{-(6) \pm \sqrt{(6)^2 - 4(2)(-3)}}{2(2)} = \frac{-6 \pm \sqrt{36 + 24}}{4}$.
$x = \frac{-6 \pm \sqrt{60}}{4} = \frac{-6 \pm 2\sqrt{15}}{4} = \frac{-3 \pm \sqrt{15}}{2}$.

Think of the quadratic formula as the ultimate cheat code for solving these types of equations. Instead of wrestling with factoring or completing the square every single time, you can just plug in your $a$, $b$, and $c$ values. It’s a powerful shortcut that works for any quadratic equation you encounter.

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A perfect square trinomial comes from squaring a binomial, like $(x + n)^2 = x^2 + 2nx + n^2$. The key relationship is that the constant term, $c$, is the square of half the coefficient of the $x$ term, $b$. In other words, the magic formula is $c = (\frac{b}{2})^2$.

In $x^2 + 8x + 16$, half of $b=8$ is $4$, and $4^2=16$. This is a perfect square: $(x+4)^2$.
In $x^2 + 10x + 25$, half of $b=10$ is $5$, and $5^2=25$. This is a perfect square: $(x+5)^2$.
In $x^2 - 12x + 36$, half of $b=-12$ is $-6$, and $(-6)^2=36$. This is a perfect square: $(x-6)^2$.

Imagine building a literal square with algebra tiles. A perfect square trinomial is an equation that forms a perfect, gap-free square. This special relationship between the $b$ and $c$ terms is the secret to making this happen, and it's the foundation for the 'completing the square' method you'll use later on.

Property

If you have an expression like $x^2 + bx$, you can turn it into a perfect square by adding a specific value. This value is always $(\frac{b}{2})^2$. To keep the equation balanced, you must add this same amount to both sides of the equals sign. This creates a solvable perfect square trinomial.

To solve $x^2 + 6x + 1 = 0$, first move the constant: $x^2 + 6x = -1$.
Add $(\frac{6}{2})^2 = 9$ to both sides: $x^2 + 6x + 9 = -1 + 9$.
Factor the left side into a perfect square and solve: $(x+3)^2 = 8$.

What happens when an equation isn't a perfect square? You make it one! 'Completing the square' is a clever trick where you find the one missing number needed to create a perfect square on one side. It’s like finding the last puzzle piece that makes the whole picture snap into place.

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The quadratic formula is derived by applying the 'completing the square' method to the general quadratic equation, $ax^2 + bx + c = 0$. By systematically isolating the variable $x$ through algebraic steps, the general solution is revealed. This process proves that the formula works for all quadratic equations.

Start with $ax^2 + bx + c = 0$ and divide by $a$: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$.
Move the constant term: $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
Add $(\frac{b}{2a})^2$ to both sides, which simplifies to $(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$, leading to the formula.

The quadratic formula isn't just some random magic spell; it's pure mathematical logic. By taking the generic form of a quadratic equation and using the 'completing the square' technique on it, you can see exactly how the formula is built, step-by-step. It’s like seeing how a master key is designed to fit any lock.