Solving Quadratic Equations

This section explores the three main methods for solving quadratic equations: graphical method, factoring, and the quadratic formula.

Definition: The roots of a quadratic equation are the x-values where the parabola intersects the x-axis.

Example: For x² - 2x - 8 = 0, factoring gives x+2$$x-4 = 0, leading to solutions x = -2 or x = 4.

Highlight: The quadratic formula x = $-b ± √(b² - 4ac)$/2a can be used when factoring is difficult.

The Discriminant and Nature of Roots

This section details how the discriminant determines the nature and number of roots in a quadratic equation.

Definition: The discriminant is the expression b² - 4ac under the square root in the quadratic formula.

Highlight: The discriminant determines three possible scenarios:

• b² - 4ac > 0: Two real and distinct roots
• b² - 4ac = 0: One repeated real root
• b² - 4ac < 0: No real roots

Example: When b² - 4ac = 0, the parabola touches the x-axis at exactly one point, indicating a repeated root.

Understanding Basic Quadratic Functions

This section introduces the fundamental concepts of quadratic functions and their graphical representations. The quadratic function f$x$ = ax² + bx + c forms the basis of all quadratic equations.

Definition: A quadratic function is any function that can be written in the form f$x$ = ax² + bx + c, where a ≠ 0.

Highlight: The coefficient 'a' determines whether the parabola opens upward (a > 0) creating a U-shape, or downward (a < 0) creating an n-shape.

Example: For the function y = kx², when given the point (2,12), we can find k by substituting: 12 = k(2²), therefore k = 3, making the equation y = 3x².

Vocabulary: The vertical line of symmetry is a line that divides the parabola into two identical halves.