Factoring Formulas and Quadratics

Questions and Answers

What is the formula for the sum/differences of squares?

a² + b² = (a + b)(a - b)

a² + b² = (a - b)(a + b)

a² - b² = (a - b)(a - b)

a² - b² = (a + b)(a - b) (correct)

What do you do with the values 'a' and 'b' in the sum/differences of squares formula?

Square root the 'a' and 'b' values.

Which of the following represents the difference of two cubes?

a³ + b³ = (a - b)(a² + ab + b²)

a³ - b³ = (a - b)(a² + ab + b²) (correct)

a³ + b³ = (a + b)(a² - ab + b²)

a³ - b³ = (a + b)(a² - b²)

What is the quadratic formula?

x = -b ± √(b² - 4ac)/2a Signup and view all the answers

What does the discriminant indicate?

The number of real solutions Signup and view all the answers

What can you conclude if the discriminant is less than zero?

There are no real solutions and there are two imaginary solutions. Signup and view all the answers

What does 'i' represent in complex numbers?

√-1 Signup and view all the answers

What is the value of i to the power of 4?

1 Signup and view all the answers

Study Notes

Factoring Formulas

Sum/Difference of Squares:
- Formula: ( a² - b² = (a + b)(a - b) )
- To use this formula, find the square roots of 'a' and 'b'.
- Example: For ( 9c² - 4d² ), it factors to ( (3c + 2d)(3c - 2d) ).
Sum/Difference of Two Cubes:
- Difference of two cubes: ( a³ - b³ = (a - b)(a² + ab + b²) )
- Steps: Factor out any monomial, then apply the cube formula.
- Example: ( 40c³ - 5d³ ) factors to ( 5(2c - d)(4c² + 2cd + d²) ) after breaking down ( 8c³ ) to ( (2c)³ ).
- Sum of two cubes: ( a³ + b³ = (a + b)(a² - ab + b²) ).

Quadratic Formula

Quadratic Formula:
- Expressed as ( x = \frac{-b ± √(b² - 4ac)}{2a} ).
- Useful for solving quadratic equations by plugging in values for a, b, and c.

Discriminant

Discriminant:
- Represents the value inside the square root of the quadratic formula.
- Conditions based on the discriminant:
  - Greater than zero: Two real solutions.
  - Equal to zero: One real solution.
  - Less than zero: No real solutions and two imaginary solutions.

Complex Numbers

Complex Numbers:
- Defined as ( i = √-1 ).
- Power properties:
  - ( i² = -1 )
  - ( i³ = -i )
  - ( i⁴ = 1 )
  - ( i⁵ = i )
- Example for simplification: When dividing a number by 4, if there's a remainder, represent it using ( i⁴ ) and ( i ) multiplied by the remainder value.

Description

This quiz covers key mathematical concepts including the sum and difference of squares and cubes, along with the quadratic formula and discriminant. Master these formulas to effectively factor polynomials and solve quadratic equations. Test your knowledge with examples and practical applications!