Algebra 2 - Quadratics Flashcards

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Questions and Answers

What is the standard form of a quadratic function?

f(x) = ax² + bx

f(x) = ax² + bx + c (correct)

f(x) = ax + b

f(x) = ax

What shape does the graph of a quadratic function have?

Parabola

What is the vertex of a parabola?

The highest or lowest point

What is the formula for the x-coordinate of the vertex?

x = -b/2a Signup and view all the answers

What is the Axis of Symmetry of a Parabola?

The vertical line through the x-coordinate of the vertex Signup and view all the answers

What is the standard form of a quadratic equation?

ax² + bx + c = 0 Signup and view all the answers

What are roots in the context of quadratic equations?

Solutions of a quadratic equation Signup and view all the answers

What does factoring a polynomial involve?

Writing it as a product of factors Signup and view all the answers

The expression $a² - b²$ can be factored as (a + b)(_____).

(a - b) Signup and view all the answers

A perfect square trinomial can be expressed as (a + b)² = _____ + 2ab + b².

a² Signup and view all the answers

What does the Zero Product Property state?

ab = 0 Signup and view all the answers

What is the completing the square method used for?

Solving quadratic equations Signup and view all the answers

What is the quadratic formula?

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

What is the discriminant in a quadratic equation?

b² - 4ac Signup and view all the answers

What does it mean if b² - 4ac > 0?

2 real roots Signup and view all the answers

What does it mean if b² - 4ac = 0?

1 real solution Signup and view all the answers

What does it mean if b² - 4ac < 0?

0 real solutions Signup and view all the answers

What are the solutions of the equation x² - 9x + 20 = 0?

x = 4 and x = 5 Signup and view all the answers

What are the solutions of the equation 3x² - 16x - 35 = 0?

x = -5/3 and x = 7 Signup and view all the answers

What are the solutions of the equation 4x² - 2x = 0?

x = 0 and x = ½ Signup and view all the answers

What are the solutions of the equation 2x² = -7x + 3?

x = ½ and x = 3 Signup and view all the answers

What is the solution of the equation x² - 6x + 9 = 0?

x = 3 Signup and view all the answers

What does it mean if an equation has no solutions?

The equation cannot be satisfied by any real number. Signup and view all the answers

What are the solutions of the equation 4x² + 5x + 5/4 = 0?

x = (-5 ± √5)/8 Signup and view all the answers

What are the solutions of the equation (x/2)² - 4x - 2 = 0?

x = 4 ± 2√5 Signup and view all the answers

Study Notes

Quadratic Functions and Equations

Standard form of a quadratic function: f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0.
Standard form of a quadratic equation: ax² + bx + c = 0.

Characteristics of Parabolas

The graph of a quadratic function forms a shape called a parabola.
The vertex of a parabola indicates its highest or lowest point, depending on its orientation.
The x-coordinate of the vertex can be calculated using the formula: x = -b/2a.
The axis of symmetry is a vertical line running through the vertex's x-coordinate, dividing the parabola into two mirrored halves.

Solving Quadratic Equations

Roots are the solutions to a quadratic equation, representing the x-intercepts where the graph intersects the x-axis.
Factoring involves rewriting a quadratic polynomial into a product of factors for simpler calculations and solutions.

Factoring Techniques

The Difference of Perfect Squares formula states: a² - b² = (a + b)(a - b).
A Perfect Square Trinomial can take the form of (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b².

Properties of Roots and Solutions

The Zero Product Property indicates that if ab = 0, then either a = 0 or b = 0.
Completing the square is a method to solve quadratics by transforming the equation into the form x² + bx = c.

Quadratic Formula and Discriminant

The Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a), derived from the standard equation, allows for solving any quadratic equation.
The Discriminant is the expression b² - 4ac, which determines the number and nature of solutions:
- If b² - 4ac > 0, there are 2 real roots.
- If b² - 4ac = 0, there is 1 real solution.
- If b² - 4ac < 0, there are 0 real solutions.

Example Solutions

Example of roots: x = 4 and x = 5 corresponds to the equation x² - 9x + 20 = 0.
Example of roots: x = -5/3 and x = 7 corresponds to the equation 3x² - 16x - 35 = 0.
Example of roots: x = 0 and x = ½ corresponds to the equation 4x² - 2x = 0.
Example of roots: x = ½ and x = 3 corresponds to the equation 2x² = -7x + 3.
Example result: x = 3 corresponds to the equation x² - 6x + 9 = 0.
Example of no solutions: The equation 2x² - 4x + 3 = 0 has no real solutions.
Solution format: x = (-5 ± √5)/8 corresponds to the equation 4x² + 5x + 5/4 = 0.
Solution format: x = 4 ± 2√5 corresponds to the equation (x/2)² - 4x - 2 = 0.

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Description

Test your knowledge about the key concepts of quadratics with these flashcards. Each card covers essential terms such as the standard form of quadratic functions, the vertex of a parabola, and more. Perfect for Algebra 2 students looking to reinforce their understanding.