Theory of Equation notes for CAT

Some Rules for finding property of roots
(1) If an equation (i:e f(x)=0 ) contains all positive co-efficients of any powers of x , it has no positive roots then.
eg: x^4+3x^2+2x+6=0 has no positive roots .

(2) For an equation , if all the even powers of x have some sign coefficients and all the odd powers of x have the opposite sign coefficients , then it has no negative roots .

(3)Summarising DESCARTES RULE OF SIGNS:
For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

(4) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to be 2-3i and if there are three possible roots of the equation , we can conclude that the last root is real . This real roots could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum. (More about finding sum and products of roots next time )

Sum and product of roots
(1) For a cubic equation ax^3+bx^2+cx+d=o
sum of the roots = – b/a
sum of the product of the roots taken two at a time = c/a
product of the roots = -d/a

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0
sum of the roots = – b/a
sum of the product of the roots taken three at a time = c/a
sum of the product of the roots taken two at a time = -d/a
product of the roots = e/a

(3) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign
coefficients then the equation has no real roots in each case(except for x=0 in the second case.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root , then even 2-root(3) is a root .
(All these are very useful in finding number of positive , negative , real ,complex etc roots of an equation )

Quadratic Equation

Quadratic equations looks like: ax² + bx + c = 0
where a,b,c are real numbers, and a ≠ 0. Every quadratic equation can have 0, 1 or 2 real decidions derived by the formula:

The number D = b² - 4ac is called discriminant.
If D < 0 then the quadratic equation have no decidions. If D = 0 then the quadratic equation have 1 decidion x = - ^b/_2a. If D > 0 then the quadratic equation have 2 decidions.
Example:
If we have equation: x² + 3x - 4 = 0
a = 1, b = 3, c = -4

Parabola

The graph of a quadratic equatin is called a parabola.
If a > 0 then graph horns pointing down:

if a < 0 then graph horns pointing up:

The midpoint of any parabola is the point x = -b/2a.

Sign of Quadratic Equation
Let f(x) = ax^2 + bx + c , where a,b,c &# 949; R and a ≠ 0

Vieta's formulas

If x₁ and x₂ are the roots of the quadratic equation ax² + bx + c = 0 then:

These formulas are called Vieta's formulas.
We can find the roots x₁ and x₂ of a quadratic equation by solving the system above.

Maxima and minima
Maxima a>0  x= - b/2a  Max = -(b^2 -4ac)/4a
Minima a<0  -----------------------------------------(same)