some imp. pts abt quadratic equations.....
1) Remember the foll. points while solving a quadratic equation..
a) x2 + y2 = (x+y)2 - 2xy
b) x3 + y3 = (x+y)3 - 3xy(x+y)
c) x4 + y4 = (x2+y2)2 - 2x2y2
d) x5 + y5 = (x2 + y2)(x3 + y3) - x2y2(x+y)
e) x - y = ( (x+y)2 - 4xy)1/2
2) If ax2 + bx + c is satisfied by more than 2 values , its an identity and a = b = c = 0
3)Common roots ->
if ax2 + bx + c = 0 & px2 + qx + r = 0 have a common root(@) , then
@2/br - qc = @/pc - ar = 1/aq - bp
eliminating @ , we get
(pc - ar)2 = (br - qc)(aq - pb)..
b)The equations , x2 + ax + b = 0 and x2 + bx + a = 0 have a common root if a + b + 1 = 0
c)By common root we means only 1 root is common , but if both the roots are common , then co-eff. of like terms are propotional..
p/a = q/b = r/c (equations are same i.e on pt a))
3)The roots of ax2 + bx + c = 0 are both = in magnitude but opposite sign if b=0
4)Roots are = to zero , if b = 0 , c = 0
roots are = to infinity , if b = 0 , a = 0
5)Roots are opposite in sign , if a and c are of opp. sign and roots are reciprocal of each other if a = c...
6)Both roots are +ive , if a and c have same sign and opp. to that of b
7) a)An equation of degree n has n roots , real or imaginary...
b)Surds and imaginary roots always occurs in pairs , i.e if p + iq is a root then p - iq is also a root
c)An odd degree equation has atleast 1 real root , whose sign is opp. to that of its last term provided that coeff of highest degree term is +ive
d)Every equation of an even degree whose constant term is -ive and highest degree term is +ive has atleast 2 real roots , 1 +ive and 1 -ive
e)If all terms of an equation are +ive and the equation involves odd no. of powers of x , then its all roots are complex...
8)If a > 0 , then the min. value of ax2 + bx + c is (4ac - b2)/ 4a
b)If a < 0 , then max. value of ax2 + bx + c is (4ac - b2)/4a
9)The condition that 1 root of ax2 + bx + c is m times the other...is
mb2 = ac (m+1)2
10)If 1 root of ax2 + bx + c is square of another , then
b3 + a2c + ac2 = 3abc
11)If the roots of the equation ax2 + bx + c are in ratio m : n , then
mnb2 = (m+n)2 ac
self written.....m sure (this tym) that it is helpful....
still hope it helps u....
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