TIPS FOR QUADRATIC....

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)

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....