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Hi . Please I need help with these questions :

See image for question.
Answer no 5 and 6.

Hi . Please I need help with these questions : See image for question. Answer no 5 and-example-1

2 Answers

1 vote

Answer:

  • (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
User Erlesand
by
7.8k points
6 votes

Answer:

(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)
User Mallow
by
8.0k points

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