If one of the roots of the equation x^2-(p+1) x+p^2+ p-8=0...

If one of the roots of the equation $x^{2}-(p+1) x+p^{2}+$ $p-8=0$ is greater than 2 and the other root is smaller than 2 , then $p$ is such that
(A) $-\frac{11}{3}<p<3$
(B) $-2<p<3$
(C) $2<p<3$
(D) None of these

Key Concepts

Sign Analysis for Function Values

Evaluating the value of a quadratic function at a specific point can provide information about the relative positions of the roots. For a quadratic with a positive leading coefficient, if the function value at a certain point is negative, it indicates that this point lies between the two real roots, which is a key observation when determining conditions on the roots.

Quadratic Equations

A quadratic equation is a second?degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are coefficients and a is nonzero. It forms the basis for studying the behavior of parabolic graphs, and its roots can be found using methods such as the quadratic formula, factoring, or completing the square.

Relationship Between Roots and Coefficients (Vieta’s Formulas)

Vieta’s formulas establish a connection between the roots of a quadratic equation and its coefficients by stating that the sum of the roots equals -b/a and the product equals c/a. This concept is essential in relating the behavior of the roots (such as their location relative to a number) directly to the parameters in the equation.

Parameter Constraints in Polynomial Equations

When a polynomial equation contains parameters, setting up inequalities based on conditions imposed on the roots (such as one being greater and one being lesser than a specific number) allows one to solve for these parameters. This involves forming inequalities by substituting the specific value into the polynomial and solving for the parameter to ensure the desired behavior of the roots.

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