Question
Find a quadratic polynomial whose zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$.
✓
Answer
Let the zeroes of a quadratic polynomial be $a =2+\sqrt{3}$ and $b =2-\sqrt{3}$
We know that,
The quadratic polynomial with $a$ and $b$ as zeroes is $x^2-(a+b) x+a b$
Substituting values of $a$ and $b$ in the above equation, we get,
$\Rightarrow x^2-(2+\sqrt{3}+2-\sqrt{3}) x+(2+\sqrt{3})(2-\sqrt{3})$
Applying, $(a+b)(a-b)=a^2-b^2$
$\Rightarrow x^2-4 x+(4-3)$
$\Rightarrow x^2-4 x+1$
Hence, the required quadratic polynomial is
$x^2-4 x+1$
We know that,
The quadratic polynomial with $a$ and $b$ as zeroes is $x^2-(a+b) x+a b$
Substituting values of $a$ and $b$ in the above equation, we get,
$\Rightarrow x^2-(2+\sqrt{3}+2-\sqrt{3}) x+(2+\sqrt{3})(2-\sqrt{3})$
Applying, $(a+b)(a-b)=a^2-b^2$
$\Rightarrow x^2-4 x+(4-3)$
$\Rightarrow x^2-4 x+1$
Hence, the required quadratic polynomial is
$x^2-4 x+1$
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