Quadratic Equationsद्विघात समीकरणquestions and answersप्रश्न और उत्तर
In bank prelims, quadratic questions ask you to compare the roots of two equations. With factorisation and a sign rule, each takes under 30 seconds.
Unlimitedअसीमितfresh questionsनए प्रश्न
Answersउत्तरalways one tap awayहमेशा एक टैप दूर
1subtopicउप-टॉपिक
IBPS PO ~5
Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।
How quadratic equations worksद्विघात समीकरण को समझिए
Factorise by splitting the middle term: find two numbers whose product is ac and sum is b.
Sign rule: if b and c are both positive, both roots are negative; if b is negative and c positive, both are positive; if c is negative, the roots have opposite signs.
Compare every root of x with every root of y. If some x is bigger and some smaller, the relationship “cannot be established”.
Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।
Exam strategyपरीक्षा रणनीतिRarely taughtकम सिखाई जाने वाली
Back-solve from the middle option
Use it when:कब इस्तेमाल करें: The question is easy to check but slow to set up as an equation (ages, mixtures, “find the number”).
Options on this site are in ascending order (as in most printed papers), so start with B or C.
Plug it into the question. Too big? The answer is above it. Too small? Below.
At most two checks find the answer.
Exampleउदाहरण
The father is 3 times his son’s age; in 12 years he will be twice the son’s age. Son’s age? (a) 10 (b) 12 (c) 14 (d) 16
→ Try 12: father 36; in 12 years 48 and 24 → 2 times. Done.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: Checking a candidate is a few multiplications; with sorted options each check halves the choices, so two checks cover four options.
Limits:सीमाएँ: Not for “cannot be determined” options or questions with several unknowns. Write the equation then.
Option killerविकल्प-छँटनी✓ Checked on 306 cases✓ 306 मामलों पर जाँची गई
Root signs from the signs of b and c
Use it when:कब इस्तेमाल करें: Bank-exam “compare x and y” questions: often the signs alone decide the answer.
For x² + bx + c = 0 with real roots:
b +, c + → both roots negative. b −, c + → both positive. c − → one positive, one negative.
If all x-roots are negative and all y-roots positive, then x < y without solving anything.
Exampleउदाहरण
I. x² + 7x + 12 = 0 II. y² − 5y + 6 = 0
→ x-roots both negative, y-roots both positive → x < y
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: The roots multiply to c and add to −b. A positive product means equal signs, and the sum then has the sign of −b.
Limits:सीमाएँ: Needs real roots (check b² ≥ 4c). When both equations have roots of the same sign, you still have to solve them.
Solved examplesहल किए गए उदाहरण
Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।
Example 1उदाहरण 1 · Quadratic equations: compare x and y (bank exams) · Easyआसान
I. x² − 16x + 63 = 0 II. y² − 12y + 35 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x ≥ y
I: x = 9, 7.
II: y = 5, 7.
Comparing: no x is smaller than any y, and the smallest x equals the largest y.
Answer: x ≥ y.
Example 2उदाहरण 2 · Quadratic equations: compare x and y (bank exams) · Mediumमध्यम
I. x² − 2x − 48 = 0 II. y² − 25y + 154 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x < y
I: x = 8, −6.
II: y = 11, 14.
Comparing: every value of x is less than every value of y.
Answer: x < y.
Example 3उदाहरण 3 · Quadratic equations: compare x and y (bank exams) · Hardकठिन
I. 4x² − 17x + 15 = 0 II. 4y² − 19y − 30 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x = y or relationship cannot be established
I: x = 5/4, 3.
II: y = 6, −5/4.
Comparing: the roots overlap or are all equal, so no single relation holds for every pair.
Answer: x = y or relationship cannot be established.
Example 4उदाहरण 4 · Quadratic equations: compare x and y (bank exams) · Mediumमध्यम
I. x² + 17x + 72 = 0 II. y² + 24y + 144 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x > y
I: x = −8, −9.
II: y = −12, −12.
Comparing: every value of x is greater than every value of y.
Answer: x > y.
Example 5उदाहरण 5 · Quadratic equations: compare x and y (bank exams) · Mediumमध्यम
I. x² − 12x + 32 = 0 II. y² − 25y + 156 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x < y
I: x = 8, 4.
II: y = 12, 13.
Comparing: every value of x is less than every value of y.
Answer: x < y.
Example 6उदाहरण 6 · Quadratic equations: compare x and y (bank exams) · Mediumमध्यम
I. x² + 12x + 27 = 0 II. y² + 24y + 140 = 0 Find the relation between x and y.
x > y
x < y
x ≥ y
x ≤ y
x = y or relationship cannot be established
Answer:उत्तर:x > y
I: x = −3, −9.
II: y = −14, −10.
Comparing: every value of x is greater than every value of y.
Answer: x > y.
Common mistakesआम गलतियाँ
Flipping the signs of the roots after factorising.
Choosing “x > y” when only one pair satisfies it.
Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न
What does “relationship cannot be established” mean?
Some combinations of roots give x > y and others give x < y, so no single relation holds.
How do I factorise fast?
Find two numbers with product ac and sum b; the roots are their negatives divided by a.
Is this asked in SSC?
Rarely in this format; it is a bank-exam favourite.