The values of K ' such that the roots of the equation 3 x^2+(k^2-k-2) x-17=0 are equal and oppos

step 1 So in the given question we are told to find the value of k such that the roots of the equation

The values of K ' such that the roots of the equation 3...

The values of $\mathrm{K}$ ' such that the roots of the equation $3 \mathrm{x}^{2}+\left(\mathrm{k}^{2}-\mathrm{k}-2\right) \mathrm{x}-17=0$ are equal and opposite in sign are
(a) $\{-1,2\}$
(b) $\{1,2\}$
(c) $\{1,-2\}$
(d) $\{-1,-2\}$

Key Concepts

Quadratic Equation Standard Form

A quadratic equation is typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a is nonzero. Understanding this structure is fundamental because it allows one to systematically study the properties of its roots using coefficient relationships and other algebraic methods.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation are given by -b/a and c/a, respectively. These relationships, derived from Vieta's formulas, allow one to relate the coefficients of the equation to the properties of its roots without needing to solve the equation explicitly.

Condition for Roots to be Equal and Opposite in Sign

For a quadratic equation to have roots that are equal in magnitude but opposite in sign (i.e., in the form r and -r), their sum must be zero. Since the sum of the roots is -b/a, setting this to zero immediately gives a condition on the coefficient b. This is a crucial insight when determining the parameters that allow the quadratic to have the desired root symmetry.

Parameter in Quadratic Equations

When a quadratic equation contains a parameter, one studies how varying that parameter affects the equation's coefficients and, consequently, its roots. This analysis often involves setting up equations based on known conditions (such as the sum or product of the roots) to find the parameter values that result in particular root characteristics.

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