Relationship of Zeroes With Coefficients of Quadratic Polynomial


 
 
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Relationship of Zeroes With Coefficients of Quadratic Polynomial

Relationship of Zeroes With Coefficients Of Quadratic Polynomial:

If p(x)ax+b,aneq 0,;and;alpha (read as alpha) is a zero of p(x) then

                       alpha =-frac{b}{a}=frac{constant;term}{coefficient;of;x}

If p(x)=ax^{2}+bx+c, a neq 0 and alpha ,beta ( read as beta) are two zeroes of p(x) then

                      alpha + beta =frac{-b}{a}=-frac{coefficient;of;x}{coefficient;of;x^{2}}

                          alphabeta =frac{c}{a}=frac{constant;term}{coefficient;of;x^{2}}

We can use these reltaions to obtain values of several symmetric expressions involving alpha and beta in terms of a, b and c.

An expressons in alpha and beta is said to be symmetric if it remaiins unchanged when alpha and beta are intercharged. For instance, alpha ^{2}+beta ^{2},alpha ^{2}-alpha beta + beta ^{2} and frac{(alpha ^{2}+beta ^{2})}{alpha beta }  are symmetric expressions in alpha and beta.

Example:  If a and c are such that the quadratic polynomial ax^{2}-5x+c has 10 as the sum of zeroes and also as the product of zeroes, fiind a and c.

Solution:   Let alpha ,beta be the zeroes of ax^{2}-5x+c. Then

                      alpha +beta =frac{5}{a},alpha beta =frac{c}{a}

As                       10=alpha +beta =alpha beta ,  we get

                          10=frac{5}{a},10=frac{c}{a}

Rightarrow                      a=frac{5}{10}=frac{1}{2}  and c = 10a = 5

Thus,                  a=frac{1}{2},c=5

Example : If alpha and beta are zeros of the quadratic polynomial x^{2}-5x+6. find the value of alpha ^{4}+beta ^{4}.

Solution:      alpha +beta =5,alpha beta =6

 We first find alpha ^{2}+beta ^{2}. We have

                alpha ^{2}+beta ^{2}=(alpha +beta )^{2}-2alpha beta =25-2(6)=13

Next

               alpha ^{4}+beta ^{4}=(alpha ^{2}+beta ^{2})^{2}-2alpha ^{2}beta ^{2}

                               =13^{2}-2(6)^{2}

                               =169-72=97

Thus,      alpha ^{4}+beta ^{4}=97

Example: If large alpha ,beta are zeros of large p(x)=ax^{2}+bx+c then obtain quadratic polynomials whose zeros are         large -alpha ,-beta                        

Solution:  As large alpha ,beta  are zeros of large p(x)=ax^{2}+bx+c

                                      large alpha +beta =-frac{b}{a},alpha beta =frac{c}{a}

  large S=(-alpha )+(-beta )=-(alpha +beta )=frac{b}{a}

 large P=(-alpha )(-beta )=alpha beta =frac{c}{a}

Thus, the quadratic expression whose zeros are large -alpha ,-beta is

               large x^{2}-Sx+P

or         large x^{2}-frac{b}{a}x+frac{c}{a}

or         large ax^{2}-bx+c

Sample Questions
(More Questions for each concept available in Login)
Question : 1

For which value of 'a' will the polynomial large (x-4)^2+(2a-373)^2 have equal roots.

Right Option : B
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