Question

j 134 a kumon
root-coefficient relationships

1. if the roots of (x^2 - 4x + 5 = 0) are (alpha) and (\beta), evaluate the following expressions.

ex.
(alpha^2 + \beta^2)
sol (alpha + \beta = 4), (alpha\beta = 5)
(alpha^2 + \beta^2 = (alpha + \beta)^2 - 2alpha\beta = 4^2 - 2 \times 5 = 6)

(1) ((alpha - \beta)^2)
sol ((alpha - \beta)^2 =)

(2) (alpha^2\beta + alpha\beta^2)
sol (alpha^2\beta + alpha\beta^2 =)

(3) (alpha^2 + alpha\beta + \beta^2)

Explanation:

Step1: Identify root relationships

Using Vieta's formulas for \(x^2 - 4x + 5 = 0\):

$$\alpha + \beta = 4, \quad \alpha\beta = 5$$

Step2: Evaluate expression (1)

Rewrite and substitute values:

$$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = 4^2 - 4 \times 5 = 16 - 20 = -4$$

Step3: Evaluate expression (2)

Factor and substitute values:

$$\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta) = 5 \times 4 = 20$$

Step4: Evaluate expression (3)

Rewrite and substitute values:

$$\alpha^2 + \alpha\beta + \beta^2 = (\alpha + \beta)^2 - \alpha\beta = 4^2 - 5 = 16 - 5 = 11$$

Answer:

(1) \(-4\)
(2) \(20\)
(3) \(11\)