Tiêu đề XI - maths - chapter 5 - QUADRATIC EQTS _ EXPRESSIONS (52-74).pdf ✅

NARAYANAGROUP 55 QUADRATIC EQUATIONS & EXPRESSIONS JEE-MAIN-SR-MATHS VOL-I  If both roots are +ve then a,c will have the same sign different from the sign of b  If a+b+c=0 then the roots are 1 and a c  If a+c=b then the roots are -1 and a  c  If the roots are in the ratio m : n then   2 2 m  n ac  mnb (or)   2 2 m n b mn ac    If one root is square of the other then   2 2 2 a c c a b 3ac b     If one root is equal to the nth power of the other root then     1 0 1 1 1 ac   a c n  b  n n n .  If roots differ by k then 2 2 2 b ac a k   4 . W. E-13: The roots of the equation       2 a b c x b c a x c a b       0 a b c    0 are Sol: a b c b c a c a b             0  the roots are     1, c a b a b c   W.E-14: If       2 p q r x q r p x r p q       0  p q r    has equal roots the value of 2 q interms of p & q is Sol: p q r q r p r p q             0  roots are     1, r p q p q r   roots are equal      1 r p q p q r    p q r r p q         pq pr rp rq    pq rq pr p r q pr      2 2   p r 2 pr q   W.E-15: The roots of 2 ax bx c    3 0 if 3b a c   are Sol: 2 ax bx c    3 0 and 3b a c     2 ax a c x c     0; 2 ax ax cx c     0 ax x c x      1 1 0    ;  x ax c    1 0   1; c x x a      roots are 1, c a   W.E-16: If one root of the equation 2 ax bx c    0 is double the other, then the relation between a, b, c is Sol: Condition is   2 2 m n b mn ac   given m n: 1: 2    2 2 1 2 1 2 b ac     2   9 2 ac b  Transformed Equations: Let ,  be the roots of fx ax bx c 0 2     then S.No New Corresponding Roots Quadratic Equation 1 ,  f  x  0 2   1 , 1 1 f 0 x        3 k k   , k  0   0      k x f 4 , k k   k  0 f kx    0 5   k,   k f x  k  0 6 2 2  ,  f  x  0 7 3 3  ,    0 3 f x  8      1 , 1 0 1         x x f W.E-17: If  , are the roots of 2 ax bx c    0 , then the equation whose roots are 2 and 2   is Sol: f x   2 0      2       a x b x c 2 2 0   2        ax x b a a b c 4 4 2 0