Solving Quadratic Equations

Four methods. Use the one that fits the equation’s structure. Below them all sits the quadratic formula, which always works — and which is itself just completing the square performed once and saved as a recipe.

Method 1 — Square root property

Use when the variable appears only inside one squared expression.

The $\pm$ is non-negotiable. Both roots are valid (unless context excludes negatives — e.g., a length).

Method 2 — Factoring + zero-product property

Use when the quadratic factors cleanly.

The zero-product property: if $ab=0$, then $a=0$ or $b=0$. Since the right side is $0$, you can set each factor independently to zero.

Method 3 — Completing the square

Use when the quadratic doesn’t factor cleanly but you want to derive both roots directly. Same trick from lesson 10’s vertex-form work, now turned into a solver.

For $x^2+6x+4=0$:

Every quadratic can be solved this way. In fact, the quadratic formula is just this procedure applied symbolically.

Method 4 — Quadratic formula

Always works. Use when factoring is hard, awkward, or impossible.

Memorize this. It’s the workhorse of every algebra and pre-calculus class.

For $x^2+4x+1=0$:

Where does the formula come from?

The quadratic formula is just Method 3 applied to the general form $a{x}^2+bx+c=0$:

That’s it. The formula is completing the square performed once, in symbols, so you never have to redo it. This is why it always works.

The discriminant

$\Delta =b^2-4ac$ predicts the number and type of roots before you compute them:

If $\Delta$ is a perfect square, the real roots are rational; otherwise they’re irrational.

Complex roots

When $\Delta <0$, the square root introduces an imaginary number $i=\sqrt{-1}$:

ML mostly stays in the real numbers, but complex eigenvalues show up in spectral methods, recurrent neural network stability analysis, and the FFT-based methods used in some signal processing layers.

ML connection — finding stationary points

Gradient descent finds where $\nabla L(\theta )=0$ — the roots of the gradient. For convex losses (like MSE in linear regression), this is one quadratic equation per parameter, solvable by these methods. For non-convex losses (most neural nets), there are many roots; gradient descent might find a local minimum but miss the global one. The structural fact that quadratics can have one, two, or zero real roots generalizes directly: convex losses have one stationary point; complex landscapes have many.