quadratic

/kwə'drætik/

Thân thiện

A student graphs a quadratic equation on a chalkboard.

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.)

Học thuật

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

Từ đồng nghĩa

• quadratic polynomial
• quadratic equation