Question

Find the quadratic polynomial, the sum of whose zeroes is √2 and their product is-12. Hence, find the zeros of the polynomial.​

Answer 1

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Let the zeroes of the polynomial be α & β

Given, that

Sum of roots = α + β = -b/a = √2

product of the roots = αβ = c/a = -12

We know that

Quadratic polynomial = x² - (sum of zeroes)x + product of zeroes = 0

Now,

The Quadratic polynomial = x² - (α + β)x + (αβ)

= x² - √2x - 12

Finding zeroes of the polynomial using quadratic equation formula

${ \sf{ \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} }}$

Here,

a = 1 , b = -√2, c = -12

$\implies{ \sf{ \frac{ - ( - \sqrt{2} ) \pm \sqrt{ {( - \sqrt{2} ) }^{2} - 4(1)( - 12) } }{2(1)} }} \\ \implies{ \sf{ \dfrac{ \sqrt{2} \pm \sqrt{2 + 48} }{2} }} \\ \implies { \sf{\frac{ \sqrt{2} \pm \sqrt{50} }{2} }} \\ { \sf{ \implies \frac{ \sqrt{2} + 5 \sqrt{2} }{2} \: or \: \frac{ \sqrt{2} - 5 \sqrt{2} }{2}}} \\ { \sf{ \implies \frac{6 \sqrt{2} }{2} \: or \: \frac{ - 4 \sqrt{2} }{2} }} \\ { \sf{ \implies3 \sqrt{2} \: or \: - 2 \sqrt{2} }} \\ { \rm{ \underline{ \underline \red{hence \: the \: roots \: are}}}} \\ \large{ \rm{ \fbox \blue{ 3\sqrt{2} \: or -2\sqrt{2} }}}$