Question

Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
(vii) $f(x)=x^{2}-(\sqrt{3}+1) x+\sqrt{3}$
(viii) $g(x)=a(x^{2}+1) -x(a^{2}+1)$
(ix) $h(s)=2s^{2}-(1+2\sqrt{2})s+\sqrt{2}$

Answer 1

SOLUTION :  

(vii) Given :   f (x) = x²  - (√3 + 1)x + √3

=  x²  - √3x - 1x + √3

=x(x–√3)−1(x−√3)

= (x–√3)(x−1)

To find zeroes,  put f(x) = 0

(x–√3)   = 0   (x−1) = 0

x = √3 or  x = 1

Hence, Zeroes of the polynomials are α = √3  and  β = 1

VERIFICATION :  

Sum of the zeroes = − coefficient of x / coefficient of x²

α + β = −coefficient of x / coefficient of x²

√3 + 1  =  -   -(√3 +1) / 1

√3 + 1  = √3 + 1  

Product of the zeroes = constant term/ Coefficient of x²

α β = constant term / Coefficient of x²

√3 × 1   = √3 /1

√3 = √3

Hence, the relationship between the Zeroes and  its coefficients is verified.

(viii) Given : g(x) = a[(x² + 1) -x(a² + 1)]

= ax² + a - a²x - x

= ax² −a²x - x + a

= ax(x−a) −1(x–a)

= (x- a)(ax - 1)

To find zeroes,  put g(x) = 0

(x- a) = 0  or (ax - 1) = 0

x = a   or  xa = 1, x = 1/a

Hence, Zeroes of the polynomials are α = a  and  β = 1/a

VERIFICATION :  

Sum of the zeroes = − coefficient of x / coefficient of x²

α + β = −coefficient of x / coefficient of x²

a + 1/a = - -(a² + 1)/a

(a×a + 1)/a = (a² + 1)/a

(a² + 1)/a = (a² + 1)/a

Product of the zeroes = constant term/ Coefficient of x²

α β = constant term / Coefficient of x²

a× 1/a   = a/a

1 = 1  

Hence, the relationship between the Zeroes and  its coefficients is verified.

(ix) IS IN THE ATTACHMENT  

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

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