CAT 2023 Slot 1 QA Q7 Equations

Question 7 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

The equation \(x^3+(2 r+1) x^2+(4 r-1) x+2=0\) has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of \(r\) is

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Answer: 2 Equations TITA

Solution

Since \(-2\) is a root, factor: \(x^3+(2r+1)x^2+(4r-1)x+2 = (x+2)(x^2+(2r-1)x+1)\) (compare coefficients).
The other two roots are real when the discriminant of \(x^2+(2r-1)x+1\) is \(\ge 0\): \((2r-1)^2 \ge 4\).
So \(2r-1\ge2\) or \(2r-1\le-2\), i.e. \(r\ge\frac32\) or \(r\le-\frac12\).
The smallest non-negative integer satisfying this is \(r=2\).