Quadratic Equations
A quadratic is any equation you can write as ax² + bx + c = 0 with a ≠ 0, and it is the workhorse of GRE algebra. The exam almost never asks you to grind through the quadratic formula on ugly numbers — the on-screen calculator is basic, so the test-writers keep the arithmetic clean and reward whoever spots the structure fastest. Most GRE quadratics factor over the integers, most hide a difference of squares or a perfect square, and a surprising number can be answered from the sum and product of the roots without solving at all. This chapter builds that toolkit in order: factoring first because it is fastest, then the formula and the discriminant as the reliable fallback, then the two pattern-recognition shortcuts (difference of squares, sum/product of roots) that let you skip the algebra entirely — each with worked examples, the quickest method, and the traps that quietly cost points.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Always move everything to one side so the quadratic equals zero before you factor or apply the formula — factoring a half-arranged equation is the #1 error.
- Read the discriminant b² − 4ac first: positive = two roots, zero = one, negative = no real roots. It answers many "how many solutions" questions with no solving.
- One square minus another? Factor with a² − b² = (a + b)(a − b) before touching the calculator — it collapses 51² − 49² to 100 × 2 instantly.
- If a question asks only for the sum or product of the roots, use p + q = −b/a and pq = c/a and skip solving entirely.
- When a perfect-square trinomial appears (x² ± 2kx + k²), it factors to (x ± k)² and gives a single repeated root — spot it to avoid the full formula.
- A clean, perfect-square discriminant is the exam hinting the quadratic would have factored — try factoring before grinding surds.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Dividing both sides by x (e.g. in 3x² = 12x) — this silently deletes the root x = 0. Factor instead.
- Sign slips with −b: for x² − 6x + 8 the sum of roots is +6, not −6.
- Forgetting the ± in the quadratic formula and reporting only one root.
- Treating a negative discriminant as if √(negative) were 0 — it means NO real solution, which changes Quantitative Comparison answers.
- Cancelling (x − 5) from (x² − 25)/(x − 5) without noting x ≠ 5 — the simplified form loses that excluded value.
- Assuming every quadratic has two solutions; a perfect square (D = 0) has exactly one repeated root.
📈 GRE exam insight & question patterns
Quantitative Comparison — Quantity A: 63² − 37². Quantity B: 2600. Which is greater?
Factor: (63 + 37)(63 − 37) = 100 × 26 = 2600. The quantities are equal — no calculator needed.
A question defines x as a solution of x² + 2x + 5 = 0 and asks you to compare x with 0. What is the answer?
The discriminant is 4 − 20 = −16 < 0, so there is no real x. The comparison cannot be carried out / the premise has no real solution — recognise this before plugging numbers.
Numeric Entry — the roots of x² − 13x + 40 = 0 are p and q. Enter p + q.
By sum of roots, p + q = −b = 13. (You do not need that the roots are 5 and 8.)
Multiple-answer — which of the following are solutions of x² − x − 6 = 0? (Select all: −3, −2, 2, 3)
Factor to (x − 3)(x + 2) = 0, roots 3 and −2. Select −2 and 3 only.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Quadratic Equations when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Solving a quadratic
| Standard form | ax² + bx + c = 0, a ≠ 0 |
|---|---|
| Quadratic formula | x = [ −b ± √(b² − 4ac) ] / (2a) |
| Discriminant | D = b² − 4ac |
| Zero-product rule | if (x − p)(x − q) = 0 then x = p or x = q |
Structure & shortcuts
| Sum of roots | p + q = −b/a |
|---|---|
| Product of roots | p × q = c/a |
| Difference of squares | a² − b² = (a + b)(a − b) |
| Perfect-square trinomials | (x ± k)² = x² ± 2kx + k² |
| Reconstruct from roots | x² − (sum)x + (product) = 0 |