Quadratics and Factoring: Learn the Patterns COLD

Here is a scene I watch every single week. A student sees 87² − 13² on a GRE Quant question. Instead of seeing the answer, they start multiplying. They square 87. They square 13. They subtract. Ninety seconds later they have the right number, and they feel GOOD about it.

They shouldn't. The student who knew the pattern got the same answer in about ten seconds and moved on with a minute and change in the bank.

Factoring on the GRE is not a test of how clever you are. It is a test of whether you RECOGNIZE a few shapes that show up again and again. Recognition can be trained. Being clever under a clock is something you can't count on. So we're going to train.

Why seconds matter here

On the current GRE, Quant comes in two sections: 12 questions in 21 minutes, then 15 questions in 26 minutes. Do the division and you get about a minute and three quarters per question. The test got shorter, but as we covered in why the shorter GRE didn't change your pace per question, the clock per question is just as tight as ever.

That is an AVERAGE, not a rule. Some questions will take you three minutes because they're wordy and messy. You pay for those minutes with the questions you crush in twenty seconds. Factoring questions should be crushed.

The patterns you need COLD

Two quick terms first. Factoring means rewriting something that is added or subtracted as things multiplied together. A quadratic is an expression where the biggest power is a square, like x² − 5x + 6.

Here are the shapes the GRE leans on:

1. Difference of squares: x² − y² = (x + y)(x − y)
2. Square of a sum: (x + y)² = x² + 2xy + y²
3. Square of a difference: (x − y)² = x² − 2xy + y²
4. The simple trinomial: for x² + bx + c, find two numbers that MULTIPLY to c and ADD to b. So x² − 5x + 6 = (x − 2)(x − 3), because −2 times −3 is 6 and −2 plus −3 is −5.

Now stop and read this part twice. COLD means BOTH directions. When you see (x + y)(x − y), you should see x² − y². When you see x² − y², you should see the product. The test hides these patterns in both directions and in disguise. 49 − a² is a difference of squares. So is 87² − 13². So is x⁴ − 1, which is (x² + 1)(x² − 1).

The trap with two doors

A quadratic equation usually has TWO solutions. The test knows that most people stop after finding one. That is where Quantitative Comparison questions love to set the trap.

A Quantitative Comparison question gives you two quantities, A and B, and asks how they compare. The four choices are always the same: A is greater, B is greater, they are equal, or you can't tell from the information given.

The mistakes I see every week

1. Expanding everything. You see (x + y)(x − y) and you FOIL it out term by term like it's homework. INSTEAD: Look for the shape before you touch the algebra. Give yourself three seconds to ask: is this one of the patterns?
2. Dividing both sides by x. Say you see x² = 5x. You divide by x and write x = 5. You just threw away x = 0, and the test will punish you for it. INSTEAD: Move everything to one side: x² − 5x = 0. Factor: x(x − 5) = 0. Now you get both answers, 0 and 5.
3. Doing it in your head. Signs flip, a 2 goes missing, and the wrong answer you made is waiting in the choices. INSTEAD: Write the pattern on your scratch paper every single time, then plug in. Ten seconds of notes beats a careless miss.
4. Forgetting the middle term. Writing (x + y)² = x² + y² is the most common factoring error there is. INSTEAD: Picture the square split into four pieces. Those two rectangles are the 2xy. They don't vanish because you were in a hurry.
5. Reading the patterns once and calling it done. Recognizing a formula on a study sheet is NOT the same as spotting it in disguise with the clock running. INSTEAD: Drill. Redo the same questions until you see the shape before you finish reading. You want to command a small set of questions, not have a passing acquaintance with a huge pile.

Train it, don't admire it

I'll bet that right now, at least one of those patterns feels obvious to you. Good. Obvious while reading a blog is not the same as automatic with 26 minutes on the clock and 15 questions to go. The Quant section tests algebra as part of the mix, and the test writers return to these shapes because they work as traps. Your job is to make them work for YOU instead.

Here's your challenge for today. Take a blank sheet of paper. From memory, write all four patterns, forward AND backward. Then write ten disguised versions of your own, like 64 − k², 51² − 49², or n² − 10n + 25, and factor each one. Time yourself. Anything that takes more than ten seconds to recognize goes on a list. Tomorrow, do it again without looking at today's sheet. If you can't write them cold tomorrow, you didn't learn them yet. You just met them.