Question

Hi . Please I need help with these questions :

See image for question.
Answer no 5 and 6.

Answer 1

• (i) √23

• (i) 196√23

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

Answer 2

(i) √23

(ii) 196√23

General Formulas and Concepts:

Pre-Algebra

• Order of Operations: BPEMDAS

Algebra I

• Standard Form: ax² + bx + c = 0
• Quadratic Formula:
  `x=(-b\pm√(b^2-4ac) )/(2a)`

Explanation:

Step 1: Define

Standard Form: x² - 10x + 2 = 0

Step 2: Define variables

a = 1

b = -10

c = 2

Step 3: Find roots

1. Substitute:
   `x=(10\pm√((-10)^2-4(1)(2)) )/(2(1))`
2. Exponents:
   `x=(10\pm√(100-4(1)(2)) )/(2(1))`
3. Multiply:
   `x=(10\pm√(100-8) )/(2)`
4. Subtract:
   `x=(10\pm√(92) )/(2)`
5. Simplify:
   `x=(10\pm 2√(23) )/(2)`
6. Factor:
   `x=(2(5\pm √(23)) )/(2)`
7. Divide:
   `x=5\pm √(23)`

Step 4: Define roots

α > β

α = 5 + √23

β = 5 - √23

Step 5: Evaluate

i

1. Substitute:
   `(1)/(5-√(23) ) -(1)/(5+√(23) )`
2. Subtract:
   `√(23)`

ii

1. Substitute:
   `(5+√(23))^3-(5-√(23))^3`
2. Evaluate:
   `(98√(23) +470)-(470 - 98√(23) )`
3. Subtract:
   `196√(23)`