Quadratic Equations • Topic 3 of 4

Difference of Squares

The identity a² − b² = (a + b)(a − b) is one of the highest-leverage patterns on the GRE because it turns a subtraction into a product you can cancel, estimate, or read off instantly. It solves equations like x² − 49 = 0 in one line (x = ±7), and it demolishes numeric traps: 87² − 13² is not worth the calculator — it is (87 + 13)(87 − 13) = 100 × 74 = 7400. The pattern also appears in disguise, such as x² − 2 = (x + √2)(x − √2) or inside fractions where a factor cancels top and bottom. On Quantitative Comparison the framing is often "Quantity A: 51² − 49²; Quantity B: 200" — factor to (51 + 49)(51 − 49) = 100 × 2 = 200 and the two are equal, no grinding required. Any time you see one square minus another, factor before you compute; it is almost always the intended shortcut.

✅ Solved examples

1. Solve x² − 49 = 0.
x² − 49 = (x + 7)(x − 7) = 0, so x = 7 or x = −7. (Equivalently x² = 49 ⇒ x = ±7.)
2. Evaluate 96² − 4² without heavy arithmetic.
(96 + 4)(96 − 4) = 100 × 92 = 9200.
3. Simplify (x² − 25) / (x − 5) for x ≠ 5.
x² − 25 = (x + 5)(x − 5). Cancel the (x − 5): the expression equals x + 5.
4. Quantitative Comparison. Quantity A: 105² − 95². Quantity B: 2000. Which is greater?
Factor: (105 + 95)(105 − 95) = 200 × 10 = 2000. Quantity A equals Quantity B, so the two quantities are equal.

✏️ Practice — try these, take hints as needed

1. Solve x² − 121 = 0.
Difference of squares.
(x + 11)(x − 11) = 0.
Two roots, opposite signs.
x = 11 or x = −11
2. Evaluate 78² − 22².
(a + b)(a − b).
(78 + 22)(78 − 22).
100 × 56.
5600
3. Solve 4x² − 9 = 0.
Write as (2x)² − 3².
(2x + 3)(2x − 3) = 0.
Solve each factor.
x = 3/2 or x = −3/2
4. Simplify (49 − y²) / (7 + y) for y ≠ −7.
49 − y² = (7 + y)(7 − y).
Cancel (7 + y).
What remains?
7 − y
5. If x² − y² = 40 and x + y = 10, find x − y.
x² − y² = (x + y)(x − y).
40 = 10 × (x − y).
Divide.
x − y = 4

📝 Topic test — 8 questions

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