Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively $\sqrt{2},\ \frac{1}{4}$.

Given: A quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively $\sqrt{2},\ \frac{1}{4}$.

To do: To form the quadratic polynomial with the given numbers as the sum and product of its zeroes respectively $\sqrt{2},\ \frac{1}{4}$.

Solution:

Let $\alpha$ and $\beta$ are the roots of the given quadratic polynomial.

As given, $\alpha+\beta=\sqrt{2}\ ..........\ ( i)$

$\alpha\beta=\frac{1}{4}\ .........\ ( ii)$

$\therefore$ The quadratic polynomial is:

$x^2-( \alpha+\beta)x+(\alpha\beta)=0$

$\Rightarrow x^2-\sqrt{2}x+\frac{1}{4}=0$

$\Rightarrow 4x^2-4\sqrt{2}x+1=0$

Thus, the quadratic polynomial is $4x^2-4\sqrt{2}x+1=0$.

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Updated on: 10-Oct-2022

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