Let D denote the discriminant $$b^{2}-4ac$$. Hence, depending on the sign and value of D, nature of the roots would be as follows:
D<0 and abs(D) is not a perfect square: Roots are complex and irrational. They can be represented as p+iq and p-iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational.
D < 0 and abs(D) is a perfect square: Roots are complex but rational. They can be represented as p+iq and p-iq where p and q are both rational.
D=0 : Roots are real and equal. X = -b/2a
D>0 and D is not a perfect square: Roots are conjugate surds
D>0 and D is a perfect square: Roots are real, rational and unequal
The General Quadratic equation will be in the form of a$$x^{2}$$+b$$x$$+c = 0
The values of ‘x’ satisfying the equation are called the roots of the equation.
The value of roots, p and q = $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
The sum of the roots = p+q = $$\frac{-b}{a}$$
Product of roots = p*q = $$\frac{c}{a}$$
If c and a are equal then the roots are reciprocal to each other.
If b = 0, then the roots are equal and are opposite in sign.
If $$A_{n}X^{n}$$ + $$A_{n-1}X^{n-1}$$ + ... + $$A_{1}X$$ + $$A_{0}$$ = 0, then
Sum of the roots = $$-A_{n-1}/ A_{n}$$
Sum of roots taken two at a time = $$A_{n-2}/ A_{n}$$
Sum of roots taken three at a time = $$-A_{n-3}/ A_{n}$$ and so on
Product of the roots = $$(-1)^{n}-A_{0}/ A_{n}$$
Minimum and maximum values of $$ax^{2}+bx+c=0$$ :
If a > 0: minimum value = $$\frac{4ac - b^2}{4a}$$ and occurs at x = $$\frac{-b}{2a}$$
If a < 0: maximum value = $$\frac{4ac - b^2}{4a}$$ and occurs at x = $$\frac{-b}{2a}$$
Signs of the roots: Let P be product of roots and S be their sum
P>0, S>0 : Both roots are positive
P>0, S<0 : Both roots are negative
P<0, S>0 : Numerical smaller root is negative and the other root is positive
P<0, S<0 : Numerical larger root is negative and the other root is positive
Finding a quadratic equation:
If roots are given : (x-a)(x-b)=0 => $$x^2 - (a+b)x + ab = 0$$
If sum s and product p of roots are given: $$x^2 - sx + p = 0$$
If roots are reciprocals of roots of equation $$ax^2 + bx + c = 0$$, then equation is $$cx^2 + bx + a = 0$$
If roots are k more than roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y-k)^2 + b(y-k) + c = 0$$
If roots are k times roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y/k)^2 + b(y/k) + c = 0$$
Descartes Rules : A polynomial equation with n sign changes can have a maximum of n positive roots. To find the maximum possible number of negative roots, find the number of positive roots of f(-x).
An equation where highest power is odd must have at least one real root
