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Find the Zeroes of the Following Quadratic Polynomials and Verify the Relationship Between the Zeroes and the Coefficients `G(X)=A(X^2+1)-x(A^2+1)` - CBSE Class 10 - Mathematics

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Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

`g(x)=a(x^2+1)-x(a^2+1)`

Solution 1

`g(x)=a[(x^2+1)-x(a^2+1)]^2=ax^2+a-a^2x-x`

`=ax^2-[(a^2+1)-x]+0=ax^2-a^2x-x+a`

`=ax(x-a)-1(x-a)=(x-a)(ax-1)`

Zeroes of the polynomials `=1/a` and a

Sum of zeroes `=(-(a^2-1))/a`

`rArr1/a+a=(a^2+1)/a`

`rArr(a^2+1)/a=(a^2+1)/a`

Product of zeroes `=a/a`

`rArr1/axxa=a/a`

`rArr1=1`

Hence relationship verified

Solution 2

`g(x)=a[(x^2+1)-x(a^2+1)]^2=ax^2+a-a^2x-x`

`=ax^2-[(a^2+1)-x]+0=ax^2-a^2x-x+a`

`=ax(x-a)-1(x-a)=(x-a)(ax-1)`

Zeroes of the polynomials `=1/a` and a

Sum of zeroes `=(-(a^2-1))/a`

`rArr1/a+a=(a^2+1)/a`

`rArr(a^2+1)/a=(a^2+1)/a`

Product of zeroes `=a/a`

`rArr1/axxa=a/a`

`rArr1=1`

Hence relationship verified

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Solution Find the Zeroes of the Following Quadratic Polynomials and Verify the Relationship Between the Zeroes and the Coefficients `G(X)=A(X^2+1)-x(A^2+1)` Concept: Relationship Between Zeroes and Coefficients of a Polynomial.
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