# CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

###### CAT Questions | Algebra | Inequalities - Quadratic Inequalities

The question is about quadratic inequalities. Some details about roots of a quadratic equation is given and our task is to find out the range of possible values for the sum of the roots. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 9: a and b are roots of the equation x2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

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##### Method of solving this CAT Question from Algebra - Inequalities : How you would relate difference between roots with its sum?

The roots are a and b:
a + b = p and
ab = 12

(a + b)2 = p2
(a - b)2 = (a + b)2 - 4ab
=> (a - b)2 = p2 - 12 * 4 = p2 - 48

If |a - b| ≥ 12 { Difference between the roots is at least 12}
then, (a - b)2 ≥ 144

p2 - 48 ≥ 144
p2 ≥ 192
P ≥ 8 √  3  or P ≤ -8 √  3

The question is "If the difference between the roots is at least 12, what is the range of values p can take?"

##### Hence the answer is "(-∞, -2) or (-2, 2)"

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