Equations and Inequalities

  • Introduction to Equations

    An equation containing a variable is neither true nor false until a particular number is substituted for the variable. If a true statement results from such a substitutions, we say that the substitution satisfies the equation. For instance, the substitution satisfies the equation , but the substitution does not. An equation that is satisfied by […]

  • Linear Equations

    A linear equation or first-degree in is written in standard form as                                        with This solved as follows:                                                    (We subtracted from both sides)                             (We divided both sides by) In many cases, simple first-degree equations can be solved mentally. For Example, the solution of  is and                         the solution of […]

  • Applications Involving Linear Equations

    Questions that arise in the real world are usually expressed in words, rather than in mathematical symbols. For example: What will be the monthly payment on my mortgage? How much insulation must I use in my house? What course should I fly to Boston? How safe is this new product? In order to answer such […]

  • Quadratic Equations

    A quadratic equation or second degree equation in is written in standard form as with If , then quadratic equation reduces to linear equation, i.e., . The word Quadratic is derived from Latin word Quadratum which means “related to two or to make square”. Examples:             Solution of Quadratic Equations: The process of finding the […]

  • Solving Quadratic Equations by Factorisation

    The process of writing an expression as a product of two or more common factors is called method of factorization. e.g. In the above examples, are the factors of expression , are the factors of and are the factors of . While solving the quadratic equation by the method of factorization, we have the following […]

  • Solving Quadratic Equations by Completing Square

    Sometimes factorization of given quadratic equation is not possible coefficients in the quadratic equation are large numbers, then it may be difficult to factorize, so in this case, we can use completing square method. In order to solve the quadratic equation by completing square method, we have the following steps. Write equation in standard form. […]

  • Solving Quadratic Equations by Quadratic Formula

    The method of completing square is still a long method for solving the quadratic equation, so to make calculation further short and easier, a formula is developed by the mathematician to solve quadratic equation, called quadratic formula. In order to derive quadratic formula, the method of completing square is used. This is given below Write […]

  • Applications Involving Quadratic Equations

    Quadratic equations have many applications in the arts and sciences, business, economics, medicine and engineering. Example: A certain negative number added to the square of the number, the result is , what is the number? What is the positive number that fulfils this condition? Solution:             Let be a negative number             By the given […]

  • The Discriminant and Complex Roots

    The expression , which appears under the radical sign in the quadratic formula is called the discriminant of the quadratic equation .             If  , and are real numbers, you can use the algebraic sign of the discriminant to determine the number and the nature of the roots of the quadratic equation. If is positive, […]

  • Radical Equations

    An equation in which the unknown appears in a radicand is called a radical equation. For instance:     and    are radical equations. To solve a radical equation, begin by isolating the most complicated radical expression on one side of the equation, and then eliminate the radical by raising both sides of the equation to a […]