WINSEM2018-19_STS2002_SS_TT621_VL2018195001084_Reference Material I_Quadratic Equations.pdf - QUADRATIC EQUATIONS QUADRATIC EQUATIONS Introduction Basic

WINSEM2018-19_STS2002_SS_TT621_VL2018195001084_Reference Material I_Quadratic Equations.pdf

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QUADRATIC EQUATIONS Schedule a Mock Interview at facebook.com/ethnus © 2018 Ethnus Consultancy Services Pvt. Ltd. All rights reserved. This material or any portion thereof may not be reproduced or used in any manner. To report violation, ask query, or provide feedback e-mail us at [email protected] QUADRATIC EQUATIONS Introduction Basic Rules INTRODUCTION An equation of the form: 2x 2 5x + 4 = 0; we have the expression for f(x) as a quadratic expression in x. Consequently, the equation 2x 2 5x + 4 = 0 would be charaterised as a quadratic equation. This equation has exactly 2 roots (solutions) and leads to the following cases with respect to whether these roots are real/ imaginary or equal/ unequal. Case 1: Both the roots are real and equal; Case 2: Both the roots are real and unequal; Case 3: Both the roots are imaginary. A detailed discussion of quadratic equations and the analytical formula based approach to identify which of the above three cases prevails follows later in this chapter. The graph of a quadratic function is always U shaped and would just touch the X-Axis in the first case above, would cut the X-Axis twice in the second case above and would not touch the X-Axis at all in the third case above. Note that the roots or solutions of the equation are the values of ‘x’ whi ch would make the LHS of the equation equal the RHS of the equation. In other words, the equation is satisfied when the value of x becomes equal to the root of the equation. THEORY OF QUADRATIC EQUATIONS An equation of the form ax 2 + bx + c = 0 …(1)
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QUADRATIC EQUATIONS Schedule a Mock Interview at facebook.com/ethnus © 2018 Ethnus Consultancy Services Pvt. Ltd. All rights reserved. This material or any portion thereof may not be reproduced or used in any manner. To report violation, ask query, or provide feedback e-mail us at [email protected] Where a, b and c are all real and a is not equal to 0, is a quadratic equation. Then, D = (b 2 4ac) is the discriminant of the quadratic equation. If D < 0 (i.e. the discriminant is negative) then the equation has no real roots. If D > 0, (i.e. the discriminant is positive) then the equation has two distinct roots, namely, ?1 = – ? + √𝐷 2? , ?𝑛𝑑 ?2 = – ? – √𝐷 2? and then ax 2 + bx + c = a (x x 1 ) (x x 2 ) …(2) If D = 0, then the quadratic equation has equal roots given by x 1 = x 2 = b/ 2a and then ax 2 + bx + c = a (x x 1 ) 2 …(3) To represent the quadratic ax 2 + bx + c in form (2) or (3) is to expand it into linear factors. Properties of Quadratic Equations and Their Roots (i) If D is a perfect square then the roots are rational and in case it is not a perfect square then the roots are irrational. (ii) In the case of imaginary roots (D < 0) and if p + iq is one root of the quadratic equation, then the other must be the conjugate p iq and vice versa (where p and q are real and i =√ -1 ) (iii) If p + √q is one root of a quadratic equation, then the other must be the conjugate p √q and vice versa.(where p is rational and √q is a surd).
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  • Winter '17
  • sathya raja shekaran
  • Quadratic equation, Elementary algebra, Ethnus Consultancy Services Pvt

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