Algebra 2 - Quadratic Equations Flashcards

Questions and Answers

What is the standard form of a quadratic equation?

y = ax² + bx + c

What are the factors of the number 12?

1, 2, 3, 4, 6, 12 (correct)

1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12 (correct)

2, 3, 6

1, 4, 7, 12

What is the Quadratic Function?

f(x) = x²

What is a parabola?

A curved graph of a quadratic function. Signup and view all the answers

What are some real world examples of quadratic parabolas?

The Saint Louis Arch, The Golden Arches of McDonald's, The base of Eiffel Tower Signup and view all the answers

What is a vertex?

The turning point of the graph of a function. Signup and view all the answers

What is an axis of symmetry?

A vertical line that divides the parabola into two congruent halves. Signup and view all the answers

What is a domain?

A set of all x values in a function. Signup and view all the answers

What is a range?

A set of all y values in a function. Signup and view all the answers

Is this written in standard form? 3x+x²+12

False Signup and view all the answers

Is this written in standard form? 6=7x−x²

False Signup and view all the answers

Is this written in standard form? 4x²+3x+2

True Signup and view all the answers

Factor the following problem: x²+8x+12

(x+2)(x+6) Signup and view all the answers

What is a coefficient?

The number multiplied by a variable. Signup and view all the answers

What is a leading coefficient?

The first number in a line of coefficients and variables. Signup and view all the answers

What is a monomial?

An expression with just one term. Signup and view all the answers

What is a binomial?

An expression with two terms. Signup and view all the answers

What is a trinomial?

An expression with three terms. Signup and view all the answers

What is a polynomial?

An expression with more than three terms. Signup and view all the answers

What is the first step when factoring a trinomial?

Create a binomial skeleton containing the correct signs of the bx and c terms. Signup and view all the answers

What is the second step when factoring a trinomial?

Identify factors of the constant. Signup and view all the answers

What is the third step when factoring a trinomial?

Drop factors into the skeleton. Signup and view all the answers

What is the fourth step when factoring a trinomial?

Check by multiplying both binomials. Signup and view all the answers

Is the statement x² = x¹*x¹ true or false?

True Signup and view all the answers

What is the FOIL method?

Multiplying two binomial answers of a trinomial expression. Signup and view all the answers

What is the BOX method?

Multiplying two binomial answers of a trinomial expression using a 2x2 square. Signup and view all the answers

X² - 4x + 4 is an example of what?

A perfect square trinomial. Signup and view all the answers

X² - y² = (x + y)(x - y) is an example of what?

A difference of two squares. Signup and view all the answers

Define perfect square trinomials.

The product you get when you multiply a binomial by itself twice. Signup and view all the answers

Define difference of two squares.

Two terms that are squared and separated by a subtraction sign. Signup and view all the answers

What is the zero product property?

Set both factors to equal zero and solve. Signup and view all the answers

Study Notes

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.

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Description

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.