To solve for the possible values of [tex]\( k \)[/tex] in the quadratic equation [tex]\( x^2 + kx + 1 = 0 \)[/tex], given that the roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] satisfy [tex]\( \alpha^2 + \beta^2 = 27 \)[/tex], we can follow these steps:
1. Sum and Product of the Roots:
Using Vieta's formulas, for a quadratic equation of the form [tex]\( x^2 + px + q = 0 \)[/tex]:
- The sum of the roots [tex]\( \alpha + \beta \)[/tex] is given by [tex]\( -p \)[/tex], which in our case is [tex]\( -k \)[/tex].
- The product of the roots [tex]\( \alpha \beta \)[/tex] is given by [tex]\( q \)[/tex], which here is [tex]\( 1 \)[/tex].
So, we have:
[tex]\[
\alpha + \beta = -k
\][/tex]
[tex]\[
\alpha \beta = 1
\][/tex]
2. Expressing [tex]\( \alpha^2 + \beta^2 \)[/tex]:
Using the identity that relates the sum and product of the roots to the sum of their squares:
[tex]\[
\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta
\][/tex]
Substituting the known values:
[tex]\[
\alpha^2 + \beta^2 = (-k)^2 - 2(1)
\][/tex]
Simplifying this expression:
[tex]\[
\alpha^2 + \beta^2 = k^2 - 2
\][/tex]
3. Given Condition:
We are given that:
[tex]\[
\alpha^2 + \beta^2 = 27
\][/tex]
Set the equation from step 2 equal to 27:
[tex]\[
k^2 - 2 = 27
\][/tex]
4. Solving for [tex]\( k \)[/tex]:
Isolate [tex]\( k^2 \)[/tex]:
[tex]\[
k^2 - 2 = 27
\][/tex]
[tex]\[
k^2 = 29
\][/tex]
Take the square root of both sides:
[tex]\[
k = \pm \sqrt{29}
\][/tex]
Therefore, the possible values of [tex]\( k \)[/tex] are:
[tex]\[
k = \sqrt{29} \quad \text{and} \quad k = -\sqrt{29}
\][/tex]
