Find a quadratic polynomial whose zeroes are $\frac{3 - \sqrt{3}}{5} and \frac{3 + \sqrt{3}}{5}$ . The quadratic polynomial from the given two zeroes is ____.

Question

Find a quadratic polynomial whose zeroes are 3-35 and 3+35. The quadratic polynomial from the given two zeroes is ____.

Accepted Answer

Let us first know a little about quadratic equations.
Quadratic equation is an equation in one variable, having degree 2. One thing to be understood is that the number of solutions of an equation is equal to the degree of the equation.
Some important formula to remember about the quadratic equation x2+bx+c=0 are –

1) α + β = - \frac{b}{a}

2) αβ = \frac{c}{a}

We are given the two zeroes of the quadratic polynomial

\frac{3 - \sqrt{3}}{5} and \frac{3 + \sqrt{3}}{5}

. Let us assume α =

\frac{3 - \sqrt{3}}{5}

and β =

\frac{3 + \sqrt{3}}{5}

.
Now, we will find the sum and product of the two zeroes –
Sum of zeroes = α+β
⇒ α+β =

\frac{3 + \sqrt{3}}{5}

+

\frac{3 - \sqrt{3}}{5}

On simplifying we will get,
⇒ α+β =

\frac{3 + \sqrt{3} + 3 - \sqrt{3}}{5}

\Rightarrow α + β = \frac{6}{5}

Product of zeroes = αβ
⇒αβ =

\frac{3 + \sqrt{3}}{5}

×

\frac{3 - \sqrt{3}}{5}

\Rightarrow αβ = \frac{(3 - \sqrt{3}) (3 + 3 \sqrt{3})}{5 \times 5}

On simplifying we will get,

\Rightarrow αβ = \frac{9 + 3 \sqrt{3} - 3 \sqrt{3} - 3}{25}

\Rightarrow αβ = \frac{6}{25}

Now, we will put the sum and the product of the zeroes in the equation.
⇒x2−(α+β)x+αβ
Putting all the values,

\Rightarrow x^{2} - (\frac{6}{5}) x + \frac{6}{25} = 0

Simplifying the equation,

\Rightarrow \frac{x^{2} \times 25}{25} - (\frac{6 \times 5}{5 \times 5}) x + \frac{6}{25} = 0

\Rightarrow 25 x^{2} - 30 x + 6 = 0

Therefore, the quadratic polynomial from the given two zeroes is 25x2−30x+6=0.

Find a quadratic polynomial whose zeroes are $\frac{3 - \sqrt{3}}{5} and \frac{3 + \sqrt{3}}{5}$ . The quadratic polynomial from the given two zeroes is ____.

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