Quadratic Equations

Quadratic Equations

Definition

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

Types of Quadratic Equations

• Monic Quadratic Equation: A quadratic equation in which the coefficient of x^2 is 1.
• Non-Monic Quadratic Equation: A quadratic equation in which the coefficient of x^2 is not 1.

Methods of Solving Quadratic Equations

• Factorization Method: Factor the left-hand side of the equation, if possible, and equate each factor to zero to find the roots.
• Quadratic Formula: If the equation cannot be factorized, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Nature of Roots

• Real and Distinct Roots: If b^2 - 4ac > 0, the equation has two real and distinct roots.
• Real and Equal Roots: If b^2 - 4ac = 0, the equation has two real and equal roots.
• Imaginary Roots: If b^2 - 4ac < 0, the equation has two imaginary roots.

Graph of Quadratic Equations

• Parabola: The graph of a quadratic equation is a parabola that opens upwards or downwards.
• Vertex: The vertex of the parabola is the lowest or highest point, depending on whether the parabola opens upwards or downwards.
• Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola.

Applications of Quadratic Equations

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under gravity.
• Optimization Problems: Quadratic equations are used to solve optimization problems, such as finding the maximum or minimum value of a function.

Quadratic Equations

Definition

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

Types of Quadratic Equations

• A monic quadratic equation is a quadratic equation with a coefficient of x^2 equal to 1.
• A non-monic quadratic equation is a quadratic equation with a coefficient of x^2 not equal to 1.

Methods of Solving Quadratic Equations

• The factorization method involves factorizing the left-hand side of the equation, if possible, and equating each factor to zero to find the roots.
• The quadratic formula is used to solve quadratic equations that cannot be factorized: x = (-b ± √(b^2 - 4ac)) / 2a.

Nature of Roots

• If b^2 - 4ac > 0, the equation has two real and distinct roots.
• If b^2 - 4ac = 0, the equation has two real and equal roots.
• If b^2 - 4ac < 0, the equation has two imaginary roots.

Graph of Quadratic Equations

• The graph of a quadratic equation is a parabola that opens upwards or downwards.
• The vertex of the parabola is the lowest or highest point, depending on whether the parabola opens upwards or downwards.
• The axis of symmetry is the vertical line that passes through the vertex of the parabola.

Applications of Quadratic Equations

• Quadratic equations are used to model the trajectory of projectiles under gravity.
• Quadratic equations are used to solve optimization problems, such as finding the maximum or minimum value of a function.