Question

Find a quadratic polynomial if the sum and product of the zero is -1/√2 and 1/√2​

Answer 1

Solution

Given:-

$\tt \implies \: \alpha + \beta = \dfrac{ - 1}{ \sqrt{2} }$

$\tt \implies \: \alpha \beta = \dfrac{1}{ \sqrt{2} }$

The Expression of quadratic equation is

$\tt \implies \: {x}^{2} - (sum \: of \: the \: roots)x + (product \: of \: the \: roots) = 0$

It's Mean

$\tt \implies \: {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0$

Now Put the Value

$\tt \implies \: {x}^{2} - \bigg( \dfrac{ - 1}{ \sqrt{2} } \bigg)x + \dfrac{1}{ \sqrt{2} } = 0$

$\tt \implies \: {x}^{2} + \dfrac{ 1}{ \sqrt{2} } x + \dfrac{1}{ \sqrt{2} } = 0$

Now Taking Lcm

$\tt \implies \dfrac{ \sqrt{2} {x}^{2} + x + 1}{ \sqrt{2} } = 0$

$\tt \implies \: \sqrt{2} {x}^{2} + x + 1 = 0$

The quadratic polynomial Equation is

$\tt \implies \: \sqrt{2} {x}^{2} + x + 1 = 0$