Question

Find a quadratic polynomial whose zeros are 3 + root 2 and 3 minus root 2

Answer 1

GIVEN :

The zeroes are $3+\sqrt{2}$ and $3-\sqrt{2}$

TO FIND :

The quadratic polynomial for given zeroes $3+\sqrt{2}$ and $3-\sqrt{2}$

SOLUTION :

Given that the zeroes are  $3+\sqrt{2}$ and $3-\sqrt{2}$

Now we have to find a quadratic polynomial with given zeroes :

Let $\alpha$ and $\beta$ be the given zeroes  $3+\sqrt{2}$ and $3-\sqrt{2}$ respectively.

i.e.,  $\alpha=3+\sqrt{2}$ and $\beta=3-\sqrt{2}$

The formula for quadratic equation with given zeroes is given by :

$x^2-(sum of the zeroes)x+(product of the zeroes)=0$

Now $Sum of the zeroes=3+\sqrt{2}+3-\sqrt{2}$

=6

∴ Sum of the zeroes=6

$Product of the zeroes=(3+\sqrt{2})(3-\sqrt{2})$

$=3^2-(\sqrt{2})^2$

By using the Algebraic identity :

$(a+b)(a-b)=a^2-b^2$

Here a=3 and $b=\sqrt{2}$

=9-2

=7

∴ Product of the zeroes=7

Substitute the values in the formula $x^2-(sum of the zeroes)x+(product of the zeroes)=0$ we get,

$x^2-(6)x+(7)=0$

$x^2-6x+7=0$

∴ the quadratic polynomial from the given zeroes $3+\sqrt{2}$ and $3-\sqrt{2}$ is $x^2-6x+7=0$