Definition

Noun: An equation in which the highest power of the unknown variable is two (a square). It is a polynomial equation of the second degree, typically written in the standard form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants and ( a \neq 0 ).

Usage

The term "quadratic equation" is used to describe and classify this specific type of algebraic equation. It is a core concept in algebra. - Solving a quadratic equation: Finding the values of the variable (roots) that satisfy the equation. - Graph of a quadratic equation: The graph is a curve called a parabola.

Examples

• In a mathematical context:
  • The student learned to solve the quadratic equation ( x^2 - 5x + 6 = 0 ) by factoring.
  • The quadratic formula provides a general solution for any quadratic equation.
• In general discussion:
  • Understanding quadratic equations is fundamental for advanced mathematics.

Advanced Usage

• Discriminant: The expression ( b^2 - 4ac ) within the quadratic formula, which determines the nature (real or complex) and number of the equation's roots.
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
• Completing the square: A method for solving a quadratic equation by manipulating it into a perfect square trinomial.
  • One way to derive the quadratic formula is by completing the square on the general quadratic equation.

Variants and Related Words

• Quadratic (adjective): Of, relating to, or involving the second and no higher power of an unknown quantity or variable.
  • The quadratic function ( f(x) = ax^2 + bx + c ) is associated with its corresponding quadratic equation.
• Quadratic formula (noun phrase): The formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) used to find the roots of a quadratic equation.

Synonyms

• Second-degree equation: A less common synonym emphasizing the highest power of the variable.
• Equation of the second order: A formal synonym used in some mathematical contexts.

Related Phrases

• To solve a quadratic equation: The standard phrase for finding its roots.
  • You can solve a quadratic equation by graphing, factoring, or using the quadratic formula.
• Roots of a quadratic equation: Refers to the solutions of the equation.
  • The roots of the quadratic equation were found to be 2 and 3.

Noun

1. an equation in which the highest power of an unknown quantity is a square