Find the zeros of quadratic polynomials and verify the relationship be

1.	Find the zeros of quadratic polynomials and verify the relationship between the zeros and their coefficients:g(x) = a(x2 + 1) –x (a2 + 1)
Answer» Given, g(x) = a(x²+1) – x(a²+1) We put g(x) = 0 ⇒ a(x²+1)–x(a²+1) = 0 ⇒ ax² + a − a²x – x = 0 ⇒ ax² − a²x – x + a = 0 ⇒ ax(x − a) − 1(x – a) = 0 ⇒ (x – a)(ax – 1) = 0 This gives us 2 zeros, for x = a and x = 1/a Hence, the zeros of the quadratic equation are a and 1/a. Now, for verification Sum of zeros = – coefficient of x / coefficient of x² a + 1/a = – (-(a² + 1)) / a (a² + 1)/a = (a² + 1)/a Product of roots = constant / coefficient of x² a x 1/a = a / a 1 = 1 Therefore, the relationship between zeros and their coefficients is verified.

Find the zeros of quadratic polynomials and verify the relationship between the zeros and their coefficients:g(x) = a(x2 + 1) –x (a2 + 1)

Answer»

Given,

g(x) = a(x²+1) – x(a²+1)

We put g(x) = 0

⇒ a(x²+1)–x(a²+1) = 0

⇒ ax² + a − a²x – x = 0

⇒ ax² − a²x – x + a = 0

⇒ ax(x − a) − 1(x – a) = 0

⇒ (x – a)(ax – 1) = 0

This gives us 2 zeros, for

x = a and x = 1/a

Hence, the zeros of the quadratic equation are a and 1/a.

Now, for verification

Sum of zeros = – coefficient of x / coefficient of x²

a + 1/a = – (-(a² + 1)) / a

(a² + 1)/a = (a² + 1)/a

Product of roots = constant / coefficient of x²

a x 1/a = a / a

1 = 1

Therefore, the relationship between zeros and their coefficients is verified.