Algebra 2 - Quadratic Equations Flashcards

Quadratic Equations

• Standard form of a quadratic equation: ( y = ax^2 + bx + c ) (e.g., ( x^2 + 8x + 12 ))
• Non-standard example: ( x - 4 + 3x + x^4 )

Factoring

• Factoring involves breaking down numbers or expressions into products of their factors.
• Example of factors of 12: ( 1, 2, 3, 4, 6, 12 ) and their negative counterparts.

Quadratic Function

• A quadratic function is defined as ( f(x) = x^2 ).
• Example coordinate calculation: For ( x = -2 ), ( f(-2) = 4 ) leading to point (-2, 4).

Parabola

• A parabola is the graphical representation of a quadratic function.

Real-World Applications

• Examples of quadratic parabolas include the Saint Louis Arch, McDonald's Golden Arches, and the base of the Eiffel Tower.

Vertex

• The vertex is the turning point of a parabola, which can be a maximum or minimum.

Axis of Symmetry

• The axis of symmetry is a vertical line that divides a parabola into two congruent mirrors.

Domain and Range

• Domain: Set of all possible x values for a function.
• Range: Set of all possible y values for a function.

Standard Form Validity

• ( 3x + x^2 + 12 ) is not in standard form; correct form is ( x^2 + 3x + 12 ).
• ( 6 = 7x - x^2 ) is not in standard form; correct form is ( -x^2 + 7x - 6 ).
• ( 4x^2 + 3x + 2 ) is in standard form.

Factoring Example

• Factoring ( x^2 + 8x + 12 ) yields ( (x + 2)(x + 6) ).

Coefficients

• A coefficient is a number multiplied by a variable (e.g., in ( 5x ), 5 is the coefficient).
• The leading coefficient is the first number in standard form (e.g., ( 7 ) in ( 7x^3 - 2x^2 + 2x - 4 )).

Types of Expressions

• Monomial: Expression with one term (e.g., ( 3x^2 )).
• Binomial: Expression with two terms (e.g., ( 3x^2 + 2 )).
• Trinomial: Expression with three terms (e.g., ( 3x^2 - 4x + 2 )).
• Polynomial: Expression with more than three terms (e.g., ( x^4 - 2x^3 + 5x^2 + x - 1 )).

Steps for Factoring Trinomials

• First step: Create a binomial skeleton with appropriate signs (e.g., ( + )( + )).
• Second step: Identify factors of the constant and their sums to match the ( bx ) term.
• Third step: Insert factors into the skeleton.
• Fourth step: Verify by multiplying the binomials.

Exponential Properties

• Exponential multiplication rule: ( x^2 = x^1 \cdot x^1 ) is true; exponents are added (e.g., ( 5x^2 \cdot x = 5x^3 )).

Methods for Checking Factorization

• FOIL method: Used for multiplying two binomials (stands for First, Outer, Inner, Last).
• BOX method: A visual method using a square to organize terms.

Special Types of Trinomials

• Perfect square trinomial: Formed by squaring a binomial (e.g., ( (x - 2)(x - 2) = x^2 - 4x + 4 )).
• Difference of two squares: Formula ( x^2 - y^2 = (x + y)(x - y) ) (e.g., ( x^2 - 4 = (x + 2)(x - 2) )).

Zero Product Property

• When the product of two factors equals zero, set each factor to zero and solve.

Test your knowledge of quadratic equations with this set of flashcards. Learn the standard form of a quadratic equation and the process of factoring. Perfect for Algebra 2 students looking to reinforce their understanding.