If n=sqrt(2)-2 is one of the roots of the equation 2 x^2-m x+m-2=0 what is the other root?

Question

Answer 1

To find the other root of the quadratic equation

, given that one of the roots is

, we can use Vieta's formulas. According to Vieta's formulas, for a quadratic equation

, the sum of the roots is

and the product of the roots is

.For the given equation \(2x^2 - mx + (m - 2) = 0\), the sum of the roots is:

and the product of the roots is:

Given that one of the roots

, let's denote the other root as

. Using the product of the roots:

Using the sum of the roots:

From the sum of the roots equation, solve for

:

Now, substitute

from the sum into the product equation:

Simplify and solve for

:

Expanding the left side:

Simplifying further:

This equation can be solved for

, but it is complex. Instead, let's directly calculate the other root using the known root and the product formula:Since

, and

, we have:

Assuming

, check if this satisfies the product condition:

Thus, the product of the roots is indeed

, which matches the value

when

.Therefore, the other root is

.

Soru