1st Year Math Notes Chapter 4

• Our comprehensive 1st Year Math Notes Chapter 4 will ensure you're fully prepared for your exams.
• Definition of quadratic equations with examples and methods of quadratic equations with examples
• (i) By Factorization  (ii) By completing the square  (iii) By Quadratic formula.
• Four Fourth Roots of Unity and Properties of Four Fourth Roots of Unity
• Synthetic Division and Outline of the Method
• Relation between the roots and the Coefficients of a Quadratic Equation
• A quadratic equation in x is an equation that can be written in the form ax2+ bx + c = 0; where a, b and c are real numbers and a ≠ 0. Another name for a quadratic equation in x is 2nd Degree Polynomial in x.
• Solution of Quadratic Equations: There are three basic techniques for solving a quadratic equation:
• i) by factorization.
• ii) by completing squares, extracting square roots.
• iii) by applying the quadratic formula.
• By Factorization: It involves factoring the polynomial ax2 + bx + c. It makes use of the fact that if ab = 0, then a = 0 or b = 0. For example, if (x - 2) (x - 4) = 0, then either x - 2 = 0 or x - 4 = 0.
• By Completing Squares, then Extracting Square Roots: Sometimes, the quadratic polynomials are not easily factorable. For example, consider x2+ 4x - 437 = 0. It is diicult to make factors of x2 + 4x - 437. In such a case the factorization and hence the solution of quadratic equation can be found by the method of completing the square and extracting square roots.
• By Applying the Quadratic Formula: Again there are some quadratic polynomials which are not factorable at all using integral coeicients. In such a case we can always ind the solution of a quadratic equation ax2+bx+c = 0 by applying a formula known as quadratic formula. This formula is applicable for every quadratic equation. Derivation of the Quadratic Formula Standard form of quadratic equation is ax2+ bx + c = 0, a ≠ 0.
• Solution of Equations Reducible to the Quadratic Equation: There are certain types of equations, which do not look to be of degree 2, but they can be reduced to the quadratic form. We shall discuss the solutions of such ive types of the equations one by one.
• Exponential Equations: Equations, in which the variable occurs in exponent, are called exponential equations.
• Reciprocal Equations: An equation, which remains unchanged when x is replaced by is called a reciprocal equation. In such an equation the coeicients of the terms equidistant from the beginning and end are equal in magnitude. The method of solving such equations is explained through the following example:
• Radical Equations: Equations involving radical expressions of the variable are called radical equations. To solve a radical equation, we irst obtain an equation free from radicals. Every solution of radical equation is also a solution of the radical-free equation but the new equation have solutions that are not solutions of the original radical equation. Such extra solutions (roots) are called extraneous roots. The method of the solution of diferent types of radical equations is illustrated by means of the followings examples:
• Remainder Theorem: If a polynomial f(x) of degree n ≥1, n is non-negative integer is divided by x - a till no x-term exists in the remainder, then f(a) is the remainder. Proof: Suppose we divide a polynomial f(x) by x - a. Then there exists a unique quotient q(x) and a unique remainder R such that f(x) = (x - a)(qx) + R.