Definition

1. Adjective:

   • Involving the second power (square) of a variable: Describes mathematical expressions, equations, or functions where the highest power of the variable is two.
   • Relating to or resembling a square in shape: Pertaining to something that is square-like or four-sided.

2. Noun:

   • A polynomial of the second degree: An algebraic expression of the form ( ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are constants and ( a \neq 0 ).
   • An equation in which the highest power of an unknown quantity is a square: An equation that can be written in the standard form ( ax^2 + bx + c = 0 ).

Usage Examples

• Adjective:

  • The student solved the quadratic equation using the formula.
  • The architect designed a building with a quadratic footprint.

• Noun:

  • The expression ( 3x^2 - 5x + 2 ) is a quadratic.
  • We learned how to factor a quadratic in algebra class.

Advanced Usage

• "Quadratic in form": An equation that can be rearranged or substituted to resemble a standard quadratic equation.

  • The equation ( x^4 + 3x^2 - 4 = 0 ) is quadratic in form if we let ( u = x^2 ).

• "Pure quadratic": A quadratic equation of the form ( ax^2 + c = 0 ), lacking the linear (( bx )) term.

  • The equation ( 2x^2 - 8 = 0 ) is a pure quadratic.

Variants and Related Words

• Quadratic function (n): A function defined by a quadratic polynomial, ( f(x) = ax^2 + bx + c ). Its graph is a parabola.
• Quadratic formula (n): The formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) used to find the roots of a quadratic equation.
• Quadratic mean (n): Another term for the root mean square (RMS), a statistical measure.

Synonyms

• Second-degree (adj): Of the second power (e.g., a second-degree polynomial).
• Squared (adj): Raised to the power of two (often used for individual terms, not the whole expression).

Related Phrases

• Quadratic residue (n): In number theory, an integer that is a perfect square modulo a given number.
• Quadratic field (n): In mathematics, an algebraic number field of degree two over the rational numbers.

Related Idioms

(Note: As a technical mathematical term, "quadratic" is not commonly used in idiomatic expressions. Its usage is primarily confined to mathematical and scientific contexts.)

Adjective

1. of or relating to the second power
   • quadratic equation
2. of or relating to or resembling a square
   • quadratic shapes

Noun

1. a polynomial of the second degree
2. an equation in which the highest power of an unknown quantity is a square