cat 2023 Question 7

Question

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

Accepted Answer

Given that -2 is a root of the given cubic equation.

=> Dividing the given equation by (x + 2), using the Horner's method of synthetic division:

coefficient of $x^2$ is 1, and coefficient of x is (2r+1)-2 = 2r-1 and the constant term = (4r-1)-2(2r-1) = 1.

=> The quadratic obtained by dividing the cubic = $x^2 + (2r - 1)x + 1 = 0$, Since, this equation has 2 real roots => Discriminant should be greater than 0

=> $(2r - 1)^2 \gt 4 $ => 2r-1 > 2 or 2r-1 < -2 => r > 3/2 or r < -1/2.

=> Minimum possible non-negative integer value of r is 2.

Question 7.

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