Quadratic Equations

Quadratic Equations

Definition

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

Solutions

• A quadratic equation has two solutions, also called roots.
• These solutions can be real or complex numbers.

Methods for Solving Quadratic Equations

1. Factoring

• If the equation can be written in the form: (x - r)(x - s) = 0, then the solutions are x = r and x = s.
• Not all quadratic equations can be factored.

2. Quadratic Formula

• The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a.
• This formula will always give the two solutions to the equation.
• The expression inside the square root (b^2 - 4ac) is called the discriminant.

Discriminant

• If the discriminant is:
  • Positive, the equation has two distinct real solutions.
  • Zero, the equation has one repeated real solution.
  • Negative, the equation has two complex solutions.

Graphing Quadratic Equations

• The graph of a quadratic equation is a parabola that opens upward or downward.
• The x-intercepts of the graph are the solutions to the equation.
• The vertex of the parabola is the minimum or maximum point of the graph.

Quadratic Equations

Definition

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

Solutions

• A quadratic equation has two solutions, also called roots.
• These solutions can be real or complex numbers.

Methods for Solving Quadratic Equations

Factoring

• If an equation can be written in the form (x - r)(x - s) = 0, then the solutions are x = r and x = s.
• Not all quadratic equations can be factored.

Quadratic Formula

• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
• This formula always gives the two solutions to the equation.
• The expression inside the square root (b^2 - 4ac) is called the discriminant.

Discriminant

• If the discriminant is:
  • Positive, the equation has two distinct real solutions.
  • Zero, the equation has one repeated real solution.
  • Negative, the equation has two complex solutions.

Graphing Quadratic Equations

• The graph of a quadratic equation is a parabola that opens upward or downward.
• The x-intercepts of the graph are the solutions to the equation.
• The vertex of the parabola is the minimum or maximum point of the graph.

Learn about quadratic equations, their definition, and methods for solving them, including factoring and more.