which value (s) of m will (x+m)^2-2(x+m)-11=x-1 have Two different negative roots?

Question

which value (s) of m will (x+m)^2-2(x+m)-11=x-1 have Two different negative roots?

Answer 1

To find the value of

such that the equation \((x+n)^2 - 2(x+n) - 11 = x - 1\) has two different negative roots, we first need to simplify and rearrange the equation.Start by expanding the left side:

Simplify this expression:

Set it equal to the right side of the original equation:

Rearrange all terms to one side:

This is a quadratic equation in the form

, where

,

, and

.For the quadratic equation to have two different negative roots, the following conditions must be satisfied:1. The discriminant must be positive:

.2. Both roots must be negative.First, calculate the discriminant:

Simplify:

Set the discriminant greater than zero:

Solve for

:

-

Check these conditions:1.

\[ 2n > 3 \quad \Rightarrow \quad n > \frac{3}{2} \]2.

Solve

using the quadratic formula or factorization: The roots of

are found using the quadratic formula:

The critical points are

and

. Since

, we have approximately

and

. The inequality

holds for

.Combine all conditions:-

The only interval satisfying all conditions is

.Thus, the integer value of

that satisfies all conditions is

.

Question

which value (s) of m will (x+m)^2-2(x+m)-11=x-1 have Two different negative roots?

Solution

Resposta