Hi . Please I need help with these questions :
See image for question.
Answer no 5 and 6.

(i) √23

(ii) 196√23

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Standard Form: ax² + bx + c = 0Quadratic Formula:

Step-by-step explanation:

Step 1: Define

Standard Form:   x² - 10x + 2 = 0

Step 2: Define variables

a = 1

b = -10

c = 2

Step 3: Find roots

Substitute:                    Exponents:                   Multiply:                        Subtract:                       Simplify:                        Factor:                          Divide:                          

Step 4: Define roots

α > β

α = 5 + √23

β = 5 - √23

Step 5: Evaluate

i

Substitute:                    Subtract:                      

ii

Substitute:                    Evaluate:                       Subtract:                      

(i) √23(i) 196√23

Step-by-step explanation:

Given equation:

x² - 10x + 2 = 0

Roots are:

α and β

Sum of the roots:

α + β = -b/a ⇒ α + β = -(-10)/1  ⇒ α + β = 10

Product of the roots:

αβ = c/a ⇒ αβ = 2/1 ⇒ αβ = 2

Finding the following:

(i)

1/β - 1/α =(α - β)/αβ =√(α - β)² / αβ =√((α + β)² - 4αβ) / αβ = √(10² - 4*2) / 2 =√92 / 2 =2√23 / 2 =√23

(ii)

α³ - β³ =(α - β)(α² + αβ + β²) =√(α - β)²× ((α + β)² - αβ)√((α + β)² - 4αβ) × ((α + β)² - αβ) =√(10² - 4*2) × (10² -2) =√92 × 98 =2√23 × 98 =196√23