Question

Find the roots of the equation below.

x2 - 6x + 12 = 0

-3 ± √3
3 ± √3
-3 ± i√3
3 ± i√3

Answer 1

`\displaystyle x = 3 \pm i√(3)`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

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

Algebra II

• Imaginary Numbers: √-1 = i

Explanation:

Step 1: Define

x² - 6x + 12 = 0

Step 2: Identify Variables

Compare the quadratic to standard form.

x² - 6x + 12 = 0 ↔ ax² + bx + c = 0

a = 1, b = -6, c = 12

Step 3: Find Roots

1. Substitute in variables [Quadratic Formula]:
   `\displaystyle x=(-(-6)\pm√((-6)^2-4(1)(12)))/(2(1))`
2. [Quadratic Formula] Simplify:
   `\displaystyle x=(6\pm√((-6)^2-4(1)(12)))/(2(1))`
3. [Quadratic Formula] [√Radical] Evaluate exponents:
   `\displaystyle x=(6\pm√(36-4(1)(12)))/(2(1))`
4. [Quadratic Formula] [√Radical] Multiply:
   `\displaystyle x=(6\pm√(36-48))/(2(1))`
5. [Quadratic Formula] [√Radical] Subtract:
   `\displaystyle x=(6\pm√(-12))/(2(1))`
6. [Quadratic Formula] [√Radical] Factor:
   `\displaystyle x=(6\pm √(-1) \cdot √(12))/(2(1))`
7. [Quadratic Formula] [√Radicals] Simplify:
   `\displaystyle x=(6 \pm 2i√(3))/(2(1))`
8. [Quadratic Formula] [Fraction - Denominator] Multiply:
   `\displaystyle x=(6 \pm 2i√(3))/(2)`
9. [Quadratic Formula] [Fraction - Numerator] Factor:
   `\displaystyle x=(2(3 \pm i√(3)))/(2)`
10. [Quadratic Formula] [Fraction] Divide:
    `\displaystyle x = 3 \pm i√(3)`