Triumphant Institute of Management Education Pvt. Ltd.(T.I.M.E.) HO:95B, 2ndFloor, Siddamsetty Complex, Secunderabad – 500 003.Tel : 040–27898195 Fax : 040–27847334 email : [email protected]website : SM1001910/8 CHAPTER – 2 QUADRATIC EQUATIONS QUADRATIC EQUATIONS “If a variable occurs in an equation with all positive integer powers and the highest power is two, then it is called a Quadratic Equation (in that variable).” In other words, a second degree polynomial in x equated to zero will be a quadratic equation. For such an equation to be a quadratic equation, the co-efficient of x² should not be zero. The most general form of a quadratic equation is ax2 + bx + c = 0, where a ≠0 (and a, b, c are real) Some examples of quadratic equations are x2– 5x + 6 = 0 .......(1) x2– x – 6 = 0 .......(2) 2x2+3x – 2 = 0 .......(3) 2x2+ x – 3 = 0 .......(4) Like a first degree equation in x has one value of x satisfying the equation, a quadratic equation in x will have TWO values of x that satisfy the equation. The values of x that satisfy the equation are called the ROOTS of the equation. These roots may be real or imaginary. For the four quadratic equations given above, the roots are as given below: Equation (1) : x = 2 and x = 3 Equation (2) : x = -2 and x = 3 Equation (3) : x = 1/2 and x = –2 Equation (4) : x = 1 and x = –3/2 In general, the roots of a quadratic equation can be found out in two ways. (i) by factorising the expression on the left-hand side of the quadratic equation (ii) by using the standard formula All the expressions may not be easy to factorise whereas applying the formula is simple and straightforward. Finding the roots by factorisation If the quadratic equation ax2+ bx + c = 0 can be written in the form (x – α)(x – β) = 0, then the roots of the equation are αand β. To find the roots of a quadratic equation, we should first write it in the form of (x – α)(x – β) = 0, i.e., the left hand side ax2+ bx + c of the quadratic equation ax2+ bx + c = 0 should be factorised into two factors. For this purpose, we should go through the following steps. We will understand these steps with the help of the equation x2– 5x + 6 = 0 which is the first of the four quadratic equations we looked at as examples above. - First write down b (the co-efficient of x) as the sum of two quantities whose product is equal to ac. In this case –5 has to be written as the sum of two quantities whose product is 6. We can write –5 as (–3) + (–2) so that the product of (–3) and (–2) is equal to 6. - Now rewrite the equation with the 'bx' term split in the above manner. In this case, the given equation can be written as x2– 3x – 2x + 6 = 0 - Take the first two terms and rewrite them together after taking out the common factor between the two of them. Similarly, the third and fourth terms should be rewritten after taking out the common factor between the two of them. In other words, you should ensure that what is left from the first and the second terms (after removing the common factor) is the same as that left from the third and the fourth terms (after removing their common factor).