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: [tex]x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
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: [tex]x=\frac{10\pm\sqrt{(-10)^2-4(1)(2)} }{2(1)}[/tex]
- Exponents: [tex]x=\frac{10\pm\sqrt{100-4(1)(2)} }{2(1)}[/tex]
- Multiply: [tex]x=\frac{10\pm\sqrt{100-8} }{2}[/tex]
- Subtract: [tex]x=\frac{10\pm\sqrt{92} }{2}[/tex]
- Simplify: [tex]x=\frac{10\pm 2\sqrt{23} }{2}[/tex]
- Factor: [tex]x=\frac{2(5\pm \sqrt{23}) }{2}[/tex]
- Divide: [tex]x=5\pm \sqrt{23}[/tex]
Step 4: Define roots
α > β
α = 5 + √23
β = 5 - √23
Step 5: Evaluate
i
- Substitute: [tex]\frac{1}{5-\sqrt{23} } -\frac{1}{5+\sqrt{23} }[/tex]
- Subtract: [tex]\sqrt{23}[/tex]
ii
- Substitute: [tex](5+\sqrt{23})^3-(5-\sqrt{23})^3[/tex]
- Evaluate: [tex](98\sqrt{23} +470)-(470 - 98\sqrt{23} )[/tex]
- Subtract: [tex]196\sqrt{23}[/tex]
