Quadratic Functions and Equations

Solving Quadratic Equations Using the Quadratic Formula

Quadratic equations are equations of the form ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula provides a universal method for finding the solutions (roots) of any quadratic equation.

• Quadratic Formula: The solutions to the equation ax2 + bx + c = 0 are given by:

• Discriminant: The expression under the square root, b2 - 4ac, is called the discriminant. It determines the nature of the roots:

  • If b2 - 4ac > 0, there are two distinct real roots.

  • If b2 - 4ac = 0, there is one real root (a repeated root).

  • If b2 - 4ac < 0, there are two complex roots.

Example Problem

Solve the equation using the quadratic formula.

• Identify coefficients: , ,

• Substitute into the quadratic formula:

• Final Answer: and

Additional info: The quadratic formula is a fundamental tool in algebra and is widely used in trigonometry for solving equations involving trigonometric identities that reduce to quadratic form.