# CAT Practice : Inequalities

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How do you find the roots of a quadratic equation?

## Inequalities - Roots

Q.22:x4 – 4x3 + ax2 – bx + 1 = 0 has positive real roots. What is the maximum possible value of a + b?
1. 20
2. 12
3. 8
4. 10

Choice D. 10

## Detailed Solution

Fabulous question.

If we have an equation of the form ax4 + bx3 + cx2 + dx + 3 = 0 with

Roots p, q, r and s.

We can say sum of the roots p + q + r + s = ${\frac{-b}{a}}$.

Sum of the products taken two at a time pq + pr + ps + qr + qs + rs = 𝑐${\frac{c}{a}}$.

Sum of the products taken three at a time pqr + pqs + prs + qrs = ${\frac{-d}{a}}$.

Product of the roots pqrs = 𝑒${\frac{e}{a}}$.

Note 1: We alternate between - and +

Note 2: Start with the immediate lower power for sum of roots, stick a

negative symbol and then alternate.

So, if p, q, r, s were roots of this equation

P + q + r + x = 4,

Pqrs = 1

Or, Arithmetic mean of p,q, r, s = 1 and geometric mean of pqrs = 1.

P, q, r s are positive real numbers. AM = GM. What does this mean?

This means that all 4 numbers are equal.

Or, this expression is (x-1)4

a = pq + pr + ps + qr + qs + rs = 6

-(-b) = pqr + pqs + prs + qrs = 4

A = 6, b = 4. A + b = 10

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## More questions from Inequalities

Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.