Algebra 2 - Quadratics Flashcards

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.