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

$h (s) = 2 s^{2} - (1 + 2 \sqrt{2}) s + \sqrt{2}$

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\sqrt{2}, \frac{1}{2}

\sqrt{3}, \frac{1}{3}

\sqrt{4}, \frac{1}{4}

\sqrt{5}, \frac{1}{5}

answer is A.

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Detailed Solution

Given, quadratic function,

h (s) = 2 s^{2} - (1 + 2 \sqrt{2}) s + \sqrt{2}

We put the value

h (s) = 0

\Rightarrow 2 s^{2} - (1 + 2 \sqrt{2}) s + \sqrt{2} = 0

\Rightarrow 2 s^{2} - 2 \sqrt{2} s - s + \sqrt{2} = 0

\Rightarrow 2 s (s - \sqrt{2}) - 1 (s - \sqrt{2}) = 0

\Rightarrow (2 s - 1) (s - \sqrt{2}) = 0

(2 s - 1) = 0 and (s - \sqrt{2}) = 0

s = \sqrt{2} and s = 1 / 2

Thus, zeros of the polynomial

s = \sqrt{2}, 1 / 2

For the verification of the relation between 0 and their coefficients
Using this formula,

Sum of zeros = - \frac{coefficient of s}{coefficient of s^{2}}

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

\frac{2 \sqrt{2} + 1}{2} = \frac{2 \sqrt{2} + 1}{2}

We calculate

Product of roots

= \frac{constant}{coefficient of s^{2}}

\frac{1}{2} \times \sqrt{2} = \frac{\sqrt{2}}{2}

\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}

Verified the relation between zeros and their coefficients.
Hence, the zeros of each of

h (s) = 2 s^{2} - (1 + 2 \sqrt{2}) s + \sqrt{2}

s = \sqrt{2}, 1 / 2

Option 1 is correct.

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