Quadratic Equation Formulas and Notes | Tips & Tricks

Some Important Result

For the quadratic equation ax^2 + bx+c =0

1. One Root will be reciprocal of other is a=c
2. One root is 0 if c=0
3. Roots are equal in magnitude but opposite in sign if b=0.
4. Both roots are zero if b=c=0.
5. Roots are positive if a and c are of same sign and b is of the opposite sign
6. Roots are opposite sign if a and c are of opposite signs.
7. Roots are negative if a,b,c are of same sign

Let f(x)= ax^2 + bx+c, where a>0.Then

1. Conditions for both the roots of f(x)=0 to be greater than a given number k are b^2 - 4ac >= 0; f(k)=0; -b/2a > k.
2. Conditions for both the roots of f(x)=0 to be less than a given number k are b^2 - 4ac >= 0; f(k)>0; -b/2a < k.
3. The number k lies between the roots of f(x)=0, if b^2 - 4ac > 0; f(k)<0.
4. Conditions for exactly one root of f(x)=0 to be lie between k1, k2 is f(k1)(k2)<0,b^2 - 4ac > 0.
5. Conditions for both roots of f(x)=0 confined between k1, k2 is f(k1)>0, f(k2)>0,  b^2 - 4ac >= 0 and k1 < -b/2a < k2, where k1<k2.
6. Conditions for both the numbers k1 and k2 lies between the roots of f(x)=0 is b^2-4ac>0; f(k1)<0; f(k2)<0.

Tips & Tricks

• An equation of degree n has n roots, real or imaginary.
• An odd degree equation has atleast one real root whose sign is opposite to that of its last term(constant term), provided that the coefficient of highest term degree is positive.
• Every equation of an even degree whose coefficient term is negative and  the coefficient of highest term degree is positive has atleast two real roots, one is positive and one is negative.
• If all the terms of an equation are positive and the equation involves no odd power of x, then its all roots are complex.
• To find the common root of two equations, make the coefficient of second degree term in the two equations equal and subtract. The value of x obtained is the required common root.