Quadratic Equations questions with solutions
Quadratic equation questions appear in almost every bank prelims as a set of five. You get two equations, one in x and one in y, and must compare the roots. With practice each question takes under 30 seconds, which makes this one of the highest-scoring topics in the paper.
How to solve quadratic equations questions
- Factorise each equation by splitting the middle term: find two numbers whose product is the constant term and whose sum is the middle coefficient.
- Change the signs of those two numbers to get the roots. For x² − 7x + 12 = 0 the numbers are −3 and −4, so x = 3 or 4.
- Compare every x with every y. If all x are bigger, x > y. If the boundaries touch (equal), use ≥ or ≤. If the ranges overlap, no relation can be established.
Common trap: Do not compare only the largest roots. One overlap is enough to make the answer “no relation”.
12 practice questions
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I. x² + 7x − 8 = 0 II. y² − 6y + 5 = 0. What is the relation between x and y?
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Answer: (D) x ≤ y. I: (x − 1)(x + 8) = 0, so x = 1 or −8. II: (y − 1)(y − 5) = 0, so y = 1 or 5. The largest x equals the smallest y and every other x is smaller, so x ≤ y.
I. x² + 17x + 72 = 0 II. y² − 13y + 40 = 0. What is the relation between x and y?
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Answer: (B) x < y. I: (x + 8)(x + 9) = 0, so x = −8 or −9. II: (y − 5)(y − 8) = 0, so y = 5 or 8. Every value of x is less than every value of y, so x < y.
I. x² − 3x + 2 = 0 II. y² − y − 20 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x − 2)(x − 1) = 0, so x = 2 or 1. II: (y + 4)(y − 5) = 0, so y = −4 or 5. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 2x − 35 = 0 II. y² − 3y − 40 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x − 7)(x + 5) = 0, so x = 7 or −5. II: (y − 8)(y + 5) = 0, so y = 8 or −5. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 5x + 6 = 0 II. y² − 7y − 18 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x − 2)(x − 3) = 0, so x = 2 or 3. II: (y − 9)(y + 2) = 0, so y = 9 or −2. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 14x + 49 = 0 II. y² + 6y − 16 = 0. What is the relation between x and y?
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Answer: (A) x > y. I: (x − 7)(x − 7) = 0, so x = 7. II: (y + 8)(y − 2) = 0, so y = −8 or 2. Every value of x is greater than every value of y, so x > y.
I. x² + x − 12 = 0 II. y² + 4y − 12 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x + 4)(x − 3) = 0, so x = −4 or 3. II: (y − 2)(y + 6) = 0, so y = 2 or −6. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² + x − 42 = 0 II. y² + y − 20 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x + 7)(x − 6) = 0, so x = −7 or 6. II: (y + 5)(y − 4) = 0, so y = −5 or 4. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² + 4x − 45 = 0 II. y² − 3y + 2 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x − 5)(x + 9) = 0, so x = 5 or −9. II: (y − 1)(y − 2) = 0, so y = 1 or 2. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 4x − 12 = 0 II. y² − 6y − 7 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x − 6)(x + 2) = 0, so x = 6 or −2. II: (y + 1)(y − 7) = 0, so y = −1 or 7. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 5x − 14 = 0 II. y² − 13y + 36 = 0. What is the relation between x and y?
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Answer: (E) x = y or no relation can be established. I: (x + 2)(x − 7) = 0, so x = −2 or 7. II: (y − 9)(y − 4) = 0, so y = 9 or 4. Some values of x are larger and some smaller than values of y, so no relation can be established.
I. x² − 4x − 5 = 0 II. y² + 11y + 28 = 0. What is the relation between x and y?
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Answer: (A) x > y. I: (x − 5)(x + 1) = 0, so x = 5 or −1. II: (y + 7)(y + 4) = 0, so y = −7 or −4. Every value of x is greater than every value of y, so x > y.
Now try it against the clock
A real mock mixes quadratic equations with everything else under a sectional timer. That is where marks are won or lost.