Quadratic Equations Properties and Forms

Quadratic Equation

Definition

• A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two.
• General form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

Properties

• Quadratic equations have two solutions, also known as roots, which can be real or complex numbers.
• The graph of a quadratic equation is a parabola that opens upward or downward.

Forms of Quadratic Equations

• Standard Form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Factored Form: (x - r)(x - s) = 0, where r and s are the roots of the equation.
• Vertex Form: a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola.

Methods for Solving Quadratic Equations

• Factoring: If the equation can be written in factored form, the roots can be easily found.
• Quadratic Formula: If the equation cannot be factored, the quadratic formula can be used: x = (-b ± √(b^2 - 4ac)) / 2a.
• Graphing: The roots of the equation can be found by graphing the related parabola and finding the x-intercepts.

Applications of Quadratic Equations

• Projectile Motion: Quadratic equations can be used to model the trajectory of projectiles, such as the path of a thrown ball.
• Optimization: Quadratic equations can be used to find the maximum or minimum value of a function, which is important in fields like economics and physics.
• Electrical Engineering: Quadratic equations are used to design and analyze electrical circuits.

Quadratic Equation

Definition

• A polynomial equation of degree two, where the highest power of the variable (usually x) is two.
• General form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

Properties

• Has two solutions, also known as roots, which can be real or complex numbers.
• Graph is a parabola that opens upward or downward.

Forms of Quadratic Equations

• Standard Form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Factored Form: (x - r)(x - s) = 0, where r and s are the roots of the equation.
• Vertex Form: a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola.

Solving Quadratic Equations

• Factoring: Find roots by factoring the equation, if possible.
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, when factoring is not possible.
• Graphing: Find roots by graphing the related parabola and finding the x-intercepts.

Applications

• Projectile Motion: Model the trajectory of projectiles, such as the path of a thrown ball.
• Optimization: Find the maximum or minimum value of a function, important in economics and physics.
• Electrical Engineering: Design and analyze electrical circuits.