Question

j 138a kumon
root-coefficient relationships

ex.
given that (x^2 - 6x + k = 0) has roots (alpha) and (alpha^2), find the value of (alpha), then find the value of (k).

sol
(\begin{cases} alpha + alpha^2 = 6 & cdots (1) \\ alpha cdot alpha^2 = alpha^3 = k & cdots (2) end{cases}) from the root-coefficient relationships of the given quadratic equation.

from (1),
(alpha^2 + alpha - 6 = 0)
((alpha + 3)(alpha - 2) = 0)
therefore, (alpha = -3, 2)

substituting into (2),
when (alpha = -3, k = -27)
when (alpha = 2, k = 8)

1. given that (x^2 - 12x + k = 0) has roots (alpha) and (alpha^2), find the value of (alpha), then find the value of (k).

Explanation:

Step1: Set up root-coefficient relationships

$$
LATEXBLOCK0
$$

Step2: Solve for \(\alpha\) using equation (1)

$$\alpha^2 + \alpha - 12 = 0$$

$$(\alpha + 4)(\alpha - 3) = 0$$

$$\alpha = -4, 3$$

Step3: Substitute \(\alpha\) into equation (2) to find \(k\)

When \(\alpha = -4\):

$$k = (-4)^3 = -64$$

When \(\alpha = 3\):

$$k = 3^3 = 27$$

Answer:

\(\alpha = -4, 3\)
When \(\alpha = -4\), \(k = -64\)
When \(\alpha = 3\), \(k = 27\)