Quadratic Equation Tricks for Bank Exams: The Short Answer
The two best quadratic equation tricks for bank exams are the sign method and the factor shortcut. The sign method reads the signs of the roots straight from the signs of the middle term and the constant, often deciding the relation between x and y without solving. The factor shortcut splits a × c into two numbers whose sum is b, then changes their signs and divides by a to get the roots in one line.
🔥 Key Exam Takeaway: In IBPS and SBI questions you only need the roots, never the full working. Find all roots of x and all roots of y, put them on one number line, and compare every x root with every y root. If even one pair disagrees, the answer is "relationship cannot be established".
Quadratic equations give 5 easy marks in almost every IBPS Clerk, SBI Clerk, IBPS PO, SBI PO, IBPS RRB and RBI Assistant prelims. Each root and answer on this page was checked in Python with exact fractions.
Quadratic Equation Questions in Bank Exams: The Format
You get two equations, one in x and one in y, and must find the relation between x and y. The usual options are:
| Option | Meaning |
|---|---|
| (a) x > y | Every x root is greater than every y root |
| (b) x ≥ y | Every x root is greater than or equal to every y root, with at least one equal pair |
| (c) x < y | Every x root is smaller than every y root |
| (d) x ≤ y | Every x root is smaller than or equal to every y root, with at least one equal pair |
| (e) x = y or relationship cannot be established | The roots overlap, or all roots are equal |
Some papers split option (e) into two separate options. Read the options before you start.
Quadratic Equation Sign Method (Sign Trick for Roots)
For ax² + bx + c = 0 with a positive, look at the signs of b and c. Sum of roots = −b/a and product of roots = c/a, so the signs of the roots follow directly.
| Sign of b | Sign of c | Signs of the roots | Example |
|---|---|---|---|
| + | + | Both negative | x² + 9x + 20 = 0 gives −4, −5 |
| − | + | Both positive | x² − 7x + 12 = 0 gives 3, 4 |
| + | − | One positive, one negative; the bigger one (in size) is negative | x² + 2x − 15 = 0 gives 3, −5 |
| − | − | One positive, one negative; the bigger one (in size) is positive | x² − x − 6 = 0 gives 3, −2 |
How to remember it: when you factorise, the roots carry the opposite signs of the numbers in the brackets. x² + 9x + 20 = (x + 4)(x + 5), so the roots are −4 and −5.
When the sign method alone gives the answer
- x has both roots negative (b and c both +) and y has both roots positive (b −, c +): x < y straight away.
- x has both roots positive and y has both roots negative: x > y straight away.
- If c is 0, one root is 0. Then use the other root to decide.
- In every other case, you need the actual roots.
Quadratic Equation Factor Shortcut (Solve in 10 Seconds)
- Multiply a × c.
- Find two numbers whose product is a × c and sum is b.
- Change the signs of both numbers.
- Divide each by a. These are the roots.
Example of the factor shortcut
2x² − 11x + 15 = 0
- a × c = 2 × 15 = 30.
- Two numbers with product 30 and sum −11: −5 and −6.
- Change signs: 5 and 6.
- Divide by a = 2: 5/2 and 3.
Check: 2 × (5/2)² − 11 × (5/2) + 15 = 12.5 − 27.5 + 15 = 0. Correct.
Tip: When a × c is large, list its factor pairs from the middle outwards. For 4x² − 25x + 36, a × c = 144 and the pair adding to 25 is 9 and 16, so the roots are 9/4 and 16/4 = 4.
How to Find the Relation Between x and y in Quadratic Equations
- Find both roots of equation I (x) and both roots of equation II (y).
- Write them on a number line, or simply in increasing order.
- Compare all four pairs (x₁ with y₁, x₁ with y₂, x₂ with y₁, x₂ with y₂).
- Decide:
- All x bigger: x > y.
- All x bigger or equal, with one tie: x ≥ y.
- All x smaller: x < y.
- All x smaller or equal, with one tie: x ≤ y.
- Mixed: relationship cannot be established (CBD).
Number line trick for comparing roots
If the x roots and y roots sit in separate blocks on the number line, the answer is > or <. If they touch at one point (a shared root at the edge), it is ≥ or ≤. If one block goes inside or crosses the other, it is CBD.
Quadratic Equation Questions with Answers (Solved Examples)
Example 1 (IBPS Clerk style)
I. x² − 7x + 12 = 0 II. y² − 9y + 20 = 0
Solution: x = 3, 4. y = 4, 5. Every x is smaller than or equal to every y, and x = 4 equals y = 4. Answer: x ≤ y.
Example 2 (sign method only)
I. x² + 9x + 20 = 0 II. y² − 3y + 2 = 0
Solution: In I, b and c are both +, so both x roots are negative. In II, b is − and c is +, so both y roots are positive. No calculation needed. (Roots: x = −4, −5 and y = 1, 2.) Answer: x < y.
Example 3 (SBI Clerk style, fractions)
I. 4x² − 25x + 36 = 0 II. 3y² − 10y + 8 = 0
Solution: For I, a × c = 144, pair −9 and −16, so x = 9/4 and 4. For II, a × c = 24, pair −4 and −6, so y = 4/3 and 2. The smallest x (9/4 = 2.25) is bigger than the largest y (2). Answer: x > y.
Example 4 (mixed signs)
I. x² + 2x − 15 = 0 II. y² − y − 6 = 0
Solution: x = 3, −5. y = 3, −2. Compare: x = −5 is smaller than y = −2, but x = 3 is bigger than y = −2. The pairs disagree. Answer: Relationship cannot be established.
Example 5 (square-root and x² = k trap)
I. x² = 144 II. y = √144
Solution: x² = 144 gives x = 12 or x = −12. But the symbol √144 means only the positive root, so y = 12. Both x values are less than or equal to 12. Answer: x ≤ y.
Special Cases in Quadratic Equations for Bank Exams
| Case | Rule | Example |
|---|---|---|
| x² = k | Two roots, +√k and −√k | x² = 196 gives x = 14, −14 |
| x = √k | Only the positive root | x = √196 gives x = 14 |
| x³ = k | One real root, same sign as k | x³ = 343 gives x = 7 |
| Perfect square | One repeated root | y² − 28y + 196 = 0 gives y = 14 |
| c = 0 | One root is 0 | x² − 5x = 0 gives x = 0, 5 |
| Shared root at the edge | Gives ≥ or ≤ | x = 3, 4 and y = 4, 5 gives x ≤ y |
Examples from this table checked in Python: x² = 196 against y² − 28y + 196 = 0 gives x ≤ y. x³ = 343 against y² = 49 gives x ≥ y.
Common Mistakes in Quadratic Equation Questions
- Forgetting the negative root of x² = k. x² = 49 means x = 7 or −7, not only 7.
- Taking both signs for √k. √49 is only 7.
- Comparing only one pair of roots. You must compare every x root with every y root.
- Missing the ≥ or ≤ case. A shared root at the edge changes > into ≥.
- Dividing by a too early. Split a × c first, then divide each number by a.
- Wrong sign after splitting. The roots are the split numbers with their signs changed.
Quick Revision Table: Quadratic Equation Tricks
| Step | What to do |
|---|---|
| Read signs of b and c | (+,+) both negative; (−,+) both positive; (+,−) bigger negative; (−,−) bigger positive |
| Factor shortcut | Split a × c into two numbers that add to b; change signs; divide by a |
| Compare | Line up all roots; separate blocks = > or <; touching = ≥ or ≤; overlap = CBD |
| x² = k | ± roots |
| √k | Positive root only |
| Target time | About 20 to 25 seconds per question |
Quadratic Equation Practice Questions for Bank Exams (with Answers)
Options: (a) x > y (b) x ≥ y (c) x < y (d) x ≤ y (e) x = y or relationship cannot be established. All answers verified in Python.
Q1. I. x² − 12x + 35 = 0 II. y² − 11y + 30 = 0 Answer: (e). x = 5, 7 and y = 5, 6. x = 5 is less than y = 6, but x = 7 is more than y = 6.
Q2. I. x² + 11x + 30 = 0 II. y² − 5y + 6 = 0 Answer: (c) x < y. Sign method: x roots both negative (−5, −6), y roots both positive (2, 3).
Q3. I. x² − 13x + 42 = 0 II. y² − 15y + 56 = 0 Answer: (d) x ≤ y. x = 6, 7 and y = 7, 8. They touch at 7.
Q4. I. 2x² − 9x + 10 = 0 II. 2y² − 13y + 21 = 0 Answer: (c) x < y. x: a × c = 20, pair −4 and −5, so x = 2, 5/2. y: a × c = 42, pair −6 and −7, so y = 3, 7/2. The largest x (2.5) is below the smallest y (3).
Q5. I. x² + 3x − 28 = 0 II. y² − 2y − 15 = 0 Answer: (e). x = 4, −7 and y = 5, −3. x = −7 is less than y = −3, but x = 4 is more than y = −3.
Q6. I. x² − x − 12 = 0 II. y² + 8y + 16 = 0 Answer: (a) x > y. x = 4, −3. y² + 8y + 16 = (y + 4)², so y = −4. Both x roots are above −4.
Q7. I. 3x² − 19x + 30 = 0 II. 2y² − 7y + 6 = 0 Answer: (a) x > y. x: a × c = 90, pair −9 and −10, so x = 3, 10/3. y: a × c = 12, pair −3 and −4, so y = 3/2, 2.
Q8. I. x² + 16x + 63 = 0 II. y² + 13y + 40 = 0 Answer: (e). x = −7, −9 and y = −5, −8. x = −7 is more than y = −8, but x = −9 is less than y = −8. The sign method alone cannot help here, because all roots are negative.
Q9. I. x² − 4x + 4 = 0 II. y² − 5y + 6 = 0 Answer: (d) x ≤ y. x = 2 (repeated root) and y = 2, 3.
Q10. I. 6x² − 13x + 6 = 0 II. 12y² − 17y + 6 = 0 Answer: (e). x: a × c = 36, pair −4 and −9, so x = 4/6 = 2/3 and 9/6 = 3/2. y: a × c = 72, pair −8 and −9, so y = 8/12 = 2/3 and 9/12 = 3/4. x = 2/3 is less than y = 3/4, but x = 3/2 is more than y = 3/4.
Quadratic Equation Tricks FAQs
How do you solve quadratic equations quickly for bank exams?
Use the factor shortcut: split a × c into two numbers that add up to b, change their signs and divide by a. With practice this takes under 10 seconds per equation.
What is the sign method in quadratic equations?
It reads the signs of the roots from the signs of b and c. If both are +, both roots are negative. If b is − and c is +, both roots are positive. If c is −, the roots have opposite signs, and the bigger one takes the opposite sign of b.
When is the answer "relationship cannot be established"?
When at least one x root is bigger than some y root and at least one x root is smaller than some y root. In other words, the x and y roots overlap on the number line.
How do you know if the answer is x ≥ y or x > y?
If every x root is greater than every y root, it is x > y. If they are all greater except for one pair that is equal, it is x ≥ y.
Does x² = 64 give one root or two?
Two: x = 8 and x = −8. But x = √64 gives only 8, because the square-root symbol means the positive root.
How many quadratic equation questions come in IBPS and SBI Clerk?
Usually a set of 5 in prelims, though some shifts use number series or approximation instead. IBPS PO and SBI PO prelims often include them too.
Related topics: revise approximation questions for bank exams, wrong number series for bank exams, probability questions for bank exams and age problems for bank exams next.
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