Quadratic Equations

The square-root method

The simplest quadratic: $(x - h)^2 = k$. If $k \ge 0$: $x - h = \pm\sqrt{k} \implies x = h \pm \sqrt{k}.$

Two solutions (or one if $k = 0$, or none if $k < 0$). Example: $(x-3)^2 = 16 \implies x-3 = \pm 4 \implies x = 7$ or $x = -1$.

Completing the square: the geometric picture

To solve $x^2 + 6x + 5 = 0$, move the constant: $x^2 + 6x = -5$.

The left side is "almost" a perfect square. Geometrically: imagine a square of side $x$ plus two rectangles of dimensions $x \times 3$ (half of 6) attached to its right and bottom. To complete the square, you need the small $3 \times 3$ corner piece:

$x^2 + 6x + 9 = -5 + 9 \implies (x+3)^2 = 4.$

We added $\left(\frac{6}{2}\right)^2 = 9$ to both sides. Then: $x+3 = \pm 2 \implies x = -1 \text{ or } x = -5.$

The key insight: you added the same number to both sides, which is legal; you chose WHICH number to add so that the left side became a perfect square. This is not magic. It is the most elegant use of the balance-beam principle in all of elementary algebra.

Deriving the quadratic formula

This is the main event. We solve the general equation $ax^2 + bx + c = 0$ with $a \ne 0$ by completing the square. Every step must be justified.

Step 1. Divide by $a$ (legal since $a \ne 0$): $x^2 + \frac{b}{a}x + \frac{c}{a} = 0.$

Step 2. Move the constant: $x^2 + \frac{b}{a}x = -\frac{c}{a}.$

Step 3. Add $\left(\frac{b}{2a}\right)^2$ to both sides: $x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}.$

Step 4. The left side is now a perfect square: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2 - 4ac}{4a^2}.$

Step 5. Apply the square-root method: $x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2|a|}.$

Since $2|a| = 2a$ when $a > 0$ and $2|a| = -2a$ when $a < 0$, and the $\pm$ absorbs the sign, we write: $x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}.$

Step 6. Subtract $\frac{b}{2a}$: $\boxed{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.}$

This is the quadratic formula. It was not handed down from the sky; it is completing the square, done once for all quadratics.

The discriminant

The expression $\Delta = b^2 - 4ac$ is the discriminant. It lives under the square root and determines the character of the solutions:

• $\Delta > 0$: two distinct real roots.
• $\Delta = 0$: exactly one real root (the vertex touches the $x$-axis).
• $\Delta < 0$: no real roots. The square root of a negative number is not real. (The solutions exist as complex numbers — but that is a story for another stratum.)

Drag the sliders for $a$, $b$, $c$ and watch the discriminant live — when the parabola dips below the $x$-axis, the discriminant turns negative. The discriminant is the algebraic detector for what you see geometrically.

When to use which method

• Factoring: use it when the polynomial factors nicely (integer or simple rational roots). Check with the discriminant: $\Delta$ is a perfect square iff factoring over $\mathbb{Z}$ is possible.
• Completing the square: use it when you want vertex form, or when understanding is required over speed.
• Quadratic formula: use it when factoring is ugly or you need exact irrational roots.

The formula always works. Factoring is faster when it works.