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4.7. NATURE OF ROOTS, , 2+ bx +c = 0 are given by, , The roots of a quadratic equation ax, _b+b* -4ac |, _ 2a |, , > 0, the roots are real and distinct and are given by, , and x =, , (i) If b? — 4ac, , , , x 2a 2a, (ii) If b? — 4ac = 0, the roots are real and equal, are given by, -b +0 i Gee, —bh -b, x= 2a or. x= 2a, , (iii) If b? — 4ac < 0, no real roots exist., , Remember, , , , , , , Discriminant Nature of roots Value of roots, , , , , , , , —b+ Vb? -4ac, 2a, ~b -b, , 2a 2a, , b? — 4ac > 0 Real and distinct, , b? — 4ac = 0 Real and equal, , b* - 4ac < 0 No real roots —, , Racine SOLVED EXAMPLES, , a 4.34. Determine the nature of roots of the following equations :, oe (ii) x? + 10x + 39 = 0, (iii) 2x? — as, oe 2 mice (iv) x? +2V3x-1=0, +Kx+1=9 (vi) 2 (a2 + bY) x2 +2 (at)?, , (vii) (x - 1) (2x — 5) =, Solution: (i) we have )=0 (viii) xv +8x+4=0, , 2x? - 3y_ 4 = 9, , , , 12! ;, , Here, . a=2,b=-3,¢=-4, Discriminant = B2 ~ dace, , oC SF ade 9» i