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Basics of Quadratic Equations

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Quadratic equations form a fundamental aspect of algebra, characterized by the standard form ax^2 + bx + c = 0. This text delves into solving techniques like factoring, completing the square, and the quadratic formula. It also discusses the discriminant's role in determining the nature of roots and the geometric representation of quadratics as parabolas. Additionally, the importance of precision in numerical solutions and the wide-ranging applications of quadratic equations in various fields are highlighted.

Exploring the Basics of Quadratic Equations

Quadratic equations are a cornerstone of algebra, defined by an equation of the form ax^2 + bx + c = 0, where 'x' represents the variable, and 'a', 'b', and 'c' are constants with 'a' not equal to zero. These constants are known as the quadratic coefficient, linear coefficient, and constant term, respectively. The solutions to a quadratic equation, also known as roots, are the values of 'x' that satisfy the equation. A quadratic equation can have two distinct solutions, one solution (a repeated root), or no real solution, with the latter case involving complex numbers.
Classic blackboard with wooden easel, parabola drawn in the centre, colored chalk, ruler and green plant in terracotta pot.

Techniques for Solving Quadratic Equations

Several methods exist for solving quadratic equations, each suited to different scenarios. Factoring, when applicable, involves rewriting the quadratic as a product of binomials. The Zero Product Property is then used to find the roots. The method of completing the square converts the equation into a perfect square trinomial, facilitating the isolation of 'x'. This technique leads to the derivation of the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), which provides a universal solution for the roots in terms of the coefficients 'a', 'b', and 'c'.

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00

The coefficients in a quadratic equation are known as the ______ coefficient, ______ coefficient, and ______ term.

quadratic

linear

constant

01

The values of 'x' that fulfill a quadratic equation are referred to as its ______.

roots

02

A quadratic equation's roots can be two distinct solutions, one solution (a ______ root), or no real solution.

repeated

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