Question

Find a quadratic polynomial whose zeroes are

1/2+ 2√3 and 1/2- 2√3

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Answer 1

Given zeroes of the polynomial are:

$= \ \textgreater \ \frac{1}{2} + 2 \sqrt{3}, \frac{1}{2} - 2 \sqrt{3}$

Now,

Sum of zeroes are:

$= \ \textgreater \ \frac{1}{2} + 2 \sqrt{3} + \frac{1}{2} - 2 \sqrt{3}$

$= \ \textgreater \ \frac{1}{2} + \frac{1}{2}$

= > 1.


Now,

Product of zeroes are:

$= \ \textgreater \ ( \frac{1}{2} + 2 \sqrt{3})( \frac{1}{2} - 2 \sqrt{3})$

we know that (a + b)(a - b) = a^2 - b^2

$= \ \textgreater \ ( \frac{1}{2})^2 - (2 \sqrt{3})^2$

$= \ \textgreater \ \frac{1}{4} - 12$

$= \ \textgreater \ \frac{1 - 48}{4}$

$= \ \textgreater \ \frac{-47}{4}$

Now,

The required Quadratic polynomial is x^2 - (sum of zeroes)x + product of zeroes.

$= \ \textgreater \ x^2 - x - ( \frac{47}{4})$


Hope this helps!