Relations between the Roots and the Coefficients of a Quadratic Equation.

How to find the relation between roots and the coefficients of a quadratic equation.

Let us the quadratic equation of the general form

ax²+bx+c=0 where a does not equal zero. a is a coefficient of x²,b is a coefficient of x, and c, is the constant term.

Now,

The standard form of a quadratic equation is

ax2+bx+c=0 where a≠0

Divided the equation by a

X²+bx/a +c/a=0

Take constant terms to the R.H.S

X²+bx/a=-c/a

To complete the square on the L.H.S add (b/2a)² to both sides.

X²+bx/a+ (b/2a)²=(b/2a)²-c/a

(x+b/2a)²=b²/4a²-c/a

(x+b/2a)²= b²-4ac/4a²

Square roots on both side

x+b/2a=±√(b^2-4ac/2a

x=-b/2a±√(b^2-4ac/2a

x=(-b±√(b^2-4ac)/2a

Hence the solution of quadratic ax²+bx+c=0 where a≠0  is given by

x=-b±√(b^2-4ac/2a

Let α and β be the roots of ax²+bx+c=0 where a≠0 such that.

α=-b+√(b^2-4ac/2a and β=-b-√(b^2-4ac/2a

Therefore,

α+β=(-b+√(b^2-4ac/2a ) + (-b-√(b^2-4ac/2a)

α+β=-b+√(b^2-4ac- b-√(b^2-4ac/2a

α+β=-2b/2a

α+β=-b/a

Sum of the roots =S=-b/a

Sum of the roots =S=-coefficient of x/coefficient of x²

Again,

αβ=(-b+√(b^2-4ac/2a)( -b-√(b^2-4ac/2a )

αβ =(-b+√(b^2-4ac)( -b-√(b^2-4ac)/4a²

αβ=(-b)2-(√(b^2-4ac)²/4a²

αβ =b²-b²+4ac/4a²

αβ=4ac/4a.a

αβ= c/a

Product of the roots =P=c/a

Product of the roots =P=constant term /coefficient of x²

The above results are helpful in expressing symmetric functions of the roots in terms of the coefficients of the quadratic equations.

Therefore, Sum of the roots (α+β =-coefficient of x/coefficient of x²) and the product of the roots (αβ=constant term/coefficient of x²) represent the required relations between roots (α andβ) and coefficients (a, b and c) of equation ax²+bx+c=0

For example:- if the roots of the equation

5x²-4x+7=0 be α and β, than

Sum of roots =-coefficient of x/coefficient of x²

Sum of roots =-(-4)/5

Sum of roots=4/5

Product of roots=constant term/coefficient of x²

Product of roots=7/5

Sum of Roots of a quadratic equation:-

The sum of the roots of a quadratic equation is equal to the negation of the coefficient of x divided by the coefficient of x².it is denoted by S.

Sum of roots =S=-b/a

Sum of roots= -coefficient of x/coefficient of x²

For example:- Sum of roots of quadratic equation

X²-5x+6=0

Compare the standard form of quadratic equation

ax2+bx+c=0 where a≠0

a=1

b=-5

c=6

Sum of roots =S=-b/a=-(-5)/1=5

Product of Roots of a quadratic equation:-

The products of the roots of a quadratic equation are equal to the constant term divided by the coefficient of x².it is denoted by P.

Product of roots =P=c/a

Product of roots =constant term /coefficient of x²

For example:-Product of roots of quadratic equation

2x²+6x+3=0

Compare the standard form of quadratic equation

ax2+bx+c=0 where a≠0

 a=2

b=6

c=3

Product of roots=c/a

Product of roots=3/2