There are many quadratics that have irrational solutions, or in some cases no real solutions at all.įor example, it is not easy at all to see how to factor the quadratic x 2 − 5 x − 3 = 0. The quadratic equations encountered so far, had one or two solutions that were rational. How many numbers must be added to make the sum equal to 152? The sum S of the first n numbers in the sequence is given by S = 2 n 2 + 3 n. We now apply this idea to solving quadratic equations.Įach number in the sequence 5, 9, 13, 17, … is obtained by adding 4 to the previous number. To factor x 2 + bx + c we try to find two numbers whose sum is b and whose product is c. In the module, Factorisation, we first saw how to factor monic quadratics, then we learnt how to factorise non-monic quadratics. The method of solving quadratic equations by factoring rests on the simple fact, used in example (2) above, that if we obtain zero as the product of two numbers then at least one of the numbers must be zero. Different situations will require different approaches, and while the last two methods always work, the method of factoring is very quick and accurate, provided the equation has rational solutions. There are three basic methods of solving such quadratic equations:Įach method is important and needs to be mastered. We will now deal with the equation ax 2 + bx + c = 0 in which neither a nor b nor c are zero. Solving quadratic equations with three terms Thus quadratic equations have been central to the history and applications of mathematics for a very long time. The techniques of solution were further refined by the Greeks, the Arabs and Indians, and finally a complete and coherent treatment was completed once the notion of complex numbers was understood. The history of quadratics will be further explored in the History section, but we note here that these types of equations were solved by both the Babylonians and Egyptians at a very early stage of world history. Surprisingly, when mathematics is employed to solve complicated and important real world problems, quadratic equations very often make an appearance as part of the overall solution. Both in senior mathematics and in tertiary and engineering mathematics, students will need to be able to solve quadratic equations with confidence and speed. While quadratic equations do not arise so obviously in everyday life, they are equally important and will frequently turn up in many areas of mathematics when more sophisticated problems are encountered. In this module we will develop a number of methods of dealing with these important types of equations. The rearrangements we used for linear equations are helpful but they are not sufficient to solve a quadratic equation. We keep rearranging the equation so that all the terms involving the unknown are on one side of the equation and all the other terms to the other side. The essential idea for solving a linear equation is to isolate the unknown. The equation = is also a quadratic equation. Thus, for example, 2 x 2 − 3 = 9, x 2 − 5 x + 6 = 0, and − 4 x = 2 x − 1 are all examples of quadratic equations. Roughly speaking, quadratic equations involve the square of the unknown. Such equations arise very naturally when solving elementary everyday problems.Ī linear equation involves the unknown quantity occurring to the first power, thus, It is based on a right triangle, and states the relationship among the lengths of the sides as \(a^2+b^2=c^2\), where \(a\) and \(b\) refer to the legs of a right triangle adjacent to the \(90°\) angle, and \(c\) refers to the hypotenuse.In the module, Linear Equations we saw how to solve various types of linear equations. One of the most famous formulas in mathematics is the Pythagorean Theorem.
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