(a+b)² = a² + 2ab + b². It's one of the most famous formulas in middle school math, but if you just memorize it, why it comes out that way might never quite click. This formula actually comes from a very natural fact: computing the area of a square two different ways gives the same answer.
Picture a square with side length (a+b). Its area is obviously (a+b)². But if you split this square by a and b along both its width and height, you get four smaller pieces: one a×a square, one b×b square, and two a×b rectangles. Add up the areas of these four pieces, and you get a² + ab + ab + b² = a² + 2ab + b², which has to equal the (a+b)² you found at the start. That's the proof that the two expressions are equal — not a formula to memorize, but a fact you can confirm with the visible quantity of area.
Now let's flip this formula backward. Take an expression like x² + 5x + 6. You can rewrite it as (x+2)(x+3) — expand it back out and you get x² + 3x + 2x + 6 = x² + 5x + 6, which checks out. Regrouping a multi-term expression like this into one made of factors multiplied together is called factoring. If the binomial formula runs in the direction of "combining split-apart pieces into one," factoring runs the opposite direction: "splitting a combined expression back apart."
Factoring is especially powerful because it's used to solve quadratic equations. Consider the equation x² + 5x + 6 = 0. Factor it, and you get (x+2)(x+3) = 0. For two numbers multiplied together to equal 0, at least one of them has to be 0 (no matter how you multiply two nonzero numbers, you can never get 0). So x+2=0 or x+3=0, and solving each gives x=−2 or x=−3. A quadratic equation that looked complicated splits, with one factoring step, into two much simpler linear equations.
The trick to factoring quickly is to use this relationship in reverse: the two roots multiplied together give the constant term, and the two roots added together with the sign flipped give the coefficient of x. In x² + bx + c = 0, find two numbers that multiply to c and add to −b, and those are exactly the two roots. For example, in x² − x − 6 = 0, the two numbers that multiply to −6 and add to 1 are 3 and −2. So (x−3)(x+2)=0, meaning x=3 or x=−2.
The binomial formula, factoring, and quadratic equations might look like separate topics on the surface, but they're really one connected story. Learn the binomial formula to expand an expression, then learn its reverse process, factoring, to regroup an expression, and suddenly a brand-new problem — the quadratic equation — can be solved with tools you already had.
On our activity page, you can change a and b with sliders to see, as square area, why (a+b)² becomes that formula, and practice guessing the two roots of a quadratic equation yourself using factoring.