f(x) = x^2 – (√3 + 1)x + √3

Question

Find the zeros of quadratic polynomials and verify the relationship between the zeros and their coefficients:

f(x) = x² – (√3 + 1)x + √3

Answer 1

Given, 

f(x) = x² – (√3 + 1)x + √3 

We put f(x) = 0 

⇒ x² – (√3 + 1)x + √3 = 0 

⇒  x² – √3x – x + √3 = 0 

⇒ x(x – √3) – 1 (x – √3) = 0 

⇒ (x – √3)(x – 1) = 0 

This gives us 2 zeros, for 

x = √3 and x = 1 

Hence, the zeros of the quadratic equation are √3 and 1. 

Now, for verification 

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

√3 + 1 = – (-(√3 +1)) / 1 

√3 + 1 = √3 +1 

Product of roots = constant / coefficient of x² 

1 x √3 = √3 / 1 

√3 = √3 

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