Definition and Properties Calculator

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Example

Created on 2024-06-20Asked by Oliver Thomas (Solvelet student)

State the definition and properties of a quadratic equation.

Solution

Definition: A quadratic equation is a polynomial equation of the form

ax^2 + bx + c = 0

, where

a \neq 0

, and

a

b

, and

c

are constants. \textbf{Properties:} 1. A quadratic equation has at most two real roots or solutions. 2. The discriminant,

\Delta = b^2 - 4ac

, determines the nature of the roots: - If

\Delta > 0

, the equation has two distinct real roots. - If

\Delta = 0

, the equation has one real root (a repeated root). - If

\Delta < 0

, the equation has two complex roots. 3. The sum of the roots of a quadratic equation is

-\frac{b}{a}

, and the product of the roots is

\frac{c}{a}

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1. Prove that the sum of two even integers is even 2. Show that the square of an odd integer is always odd.,

DefinitionA definition and Properties in mathematics is a precise statement that explains the meaning of a term, concept, or object. It gives a definite and in a strict sense meaningful description of the necessary properties of the defining entity. Definitions form the basic building blocks for the language and structure for discussions in mathematics, and they are used to define new terms and set the standards for relationships between them. A solid understanding of the definitions of mathematical objects is essential both for learning the material found in any course in mathematics, and also for developing mathematical arguments and communicating them clearly to others. For Example: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

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