Quadratic comparisons for LIC AAO

Quadratic comparison questions give you two equations, one in x and one in y, and ask how x and y compare. They look like algebra, but they are really a speed drill: factorise, list the roots, compare. Once the method is automatic, each question takes about 30 seconds and the answer is certain. That makes them some of the most reliable marks in the prelims Quant section. The errors that cost marks here are almost never about algebra. They come from comparing only some of the roots, or from the x² = k trap.

The answer options

A typical option set looks like this:

Some papers list "x = y" and "no relation" together, others separately. Choose "x = y" only when both equations have exactly one and the same root.

Factorising fast

When the coefficient of x² is 1

For x² + bx + c = 0, find two numbers that multiply to c and add to b. If they are p and q, the equation is (x + p)(x + q) = 0 and the roots are −p and −q.

For x² − 11x + 30 = 0: two numbers that multiply to 30 and add to −11 are −5 and −6. So (x − 5)(x − 6) = 0 and x = 5 or 6.

The sign rule

The signs of b and c tell you the signs of the roots before you factorise.

This works because the product of the roots is c and their sum is −b. A positive product means the roots share a sign, and the sum then shows which sign. A negative product means opposite signs.

When the coefficient of x² is not 1: split and divide

For ax² + bx + c = 0, find p and q with p × q = a × c and p + q = b. Then the roots are −p/a and −q/a. In words: change the signs of the two numbers and divide each by a.

Why it works: (ax + p)(x + q/a) expands to ax² + qx + px + pq/a. Since p + q = b and pq/a = c, this equals ax² + bx + c. Setting each factor to zero gives x = −p/a and x = −q/a.

For 2x² − 13x + 21 = 0: a × c = 42, and −6 and −7 multiply to 42 and add to −13. Change signs and divide by 2: x = 6/2 = 3 and x = 7/2 = 3.5.

Worked examples

Example 1. I. x² − 11x + 30 = 0; II. y² − 15y + 56 = 0
x = 5 or 6. For y, −7 and −8 multiply to 56 and add to −15, so y = 7 or 8.
The largest x (6) is less than the smallest y (7). x < y.

Example 2. I. 2x² − 13x + 21 = 0; II. y² − 7y + 12 = 0
From above, x = 3 or 3.5. y = 3 or 4.
Compare every pair: 3 = 3, 3 < 4, 3.5 > 3, 3.5 < 4. Some pairs have x bigger and some have y bigger. No relation.

Example 3. I. x² + 3x − 40 = 0; II. y² − 12y + 36 = 0
For x, 8 and −5 multiply to −40 and add to 3, so x = −8 or 5.
II is (y − 6)² = 0, so y = 6 only.
Both −8 and 5 are less than 6. x < y.

Example 4. I. 6x² + 5x + 1 = 0; II. 3y² + 11y + 6 = 0
For x: a × c = 6, and 2 and 3 add to 5. Roots −2/6 = −1/3 and −3/6 = −1/2.
For y: a × c = 18, and 2 and 9 add to 11. Roots −2/3 and −9/3 = −3.
On a number line, y sits at −3 and about −0.67, while x sits at −0.5 and about −0.33. Every x is greater. x > y.

Example 5. I. x² = 81; II. y = √81
x² = 81 gives x = 9 or −9. But √81 means the positive root only, so y = 9.
9 = 9 and −9 < 9. x ≤ y.

Example 6. I. 3x + 2y = 36; II. 5x + 4y = 64
This is a pair of linear equations. Multiply I by 2: 6x + 4y = 72. Subtract II: x = 8.
Then 2y = 36 − 24 = 12, so y = 6. Check in II: 40 + 24 = 64.
x > y.

The number-line habit

For every question, jot the roots on one line with x values above and y values below. Then read the answer:

• all x values to the right of all y values: x > y, or x ≥ y if one pair touches
• all x values to the left: x < y, or x ≤ y if one pair touches
• any overlap or interleaving: no relation

This single picture handles every case, including repeated roots and negatives.

Practice set

1. I. x² − 9x + 20 = 0; II. y² − 11y + 30 = 0. x ≤ y. x = 4, 5 and y = 5, 6. 5 = 5 and every other pair has x smaller.
2. I. x² + 11x + 30 = 0; II. y² + 7y + 12 = 0. x < y. x = −5, −6 and y = −3, −4. The largest x, −5, is below the smallest y, −4.
3. I. 2x² − 11x + 15 = 0; II. y² − 5y + 6 = 0. No relation. a × c = 30; −5 and −6 add to −11, so x = 2.5 or 3. y = 2 or 3. 2.5 lies between the y values.
4. I. x² − 2x − 15 = 0; II. y² + 8y + 15 = 0. x ≥ y. x = 5 or −3; y = −3 or −5. −3 = −3, and every other pair has x larger.
5. I. x² = 144; II. y² − 24y + 144 = 0. x ≤ y. x = 12 or −12; II is (y − 12)² = 0, so y = 12.
6. I. 4x² + 7x + 3 = 0; II. y² + 5y + 6 = 0. x > y. a × c = 12; 3 and 4 add to 7, so x = −3/4 or −1. y = −2 or −3. Every x is above −2.
7. I. 2x + 3y = 31; II. 3x + 2y = 29. x < y. Adding gives 5x + 5y = 60, so x + y = 12. Subtracting I from II gives x − y = −2. So x = 5 and y = 7.
8. I. x² − 13x + 42 = 0; II. y² − 13y + 40 = 0. No relation. x = 6 or 7; y = 5 or 8. The x values sit inside the y values.

A one-week drill

What to do next

• Factorise ten quadratics a day until the pairs of numbers come without thinking.
• Draw the number line for every comparison, even easy ones, for two weeks.
• Practise the other quick prelims blocks: number series and simplification and approximation.
• See where these blocks fit in a quant plan for LIC AAO.