Question

Write a quadratic polynomial whose one zero is 3-√5 and product of zeros is 4. give photo

Answer 1

as per the above image you can see that x^2 -6x +4

Answer 2

The quadratic polynomial is $\bold{x^{2}-6 x+4=0}$.

Let x,y be the two terms, such that

$\begin{array}{l}{x=3-\sqrt{5}} \\ {x \times y=4}\end{array}$

Thus, substituting the value of $x=3-\sqrt{5}$ in the above equation, we get

$\begin{array}{c}{3-\sqrt{5} \times y=4} \\ {y=\frac{4}{3-\sqrt{5}}}\end{array}$

Now, by taking conjugation of $3-\sqrt{5}$, we get  

$=\frac{4}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}}$

Now, by simplifying the above term, we get

$\begin{aligned} &=\frac{4(3+\sqrt{5})}{3^{2}(\sqrt{5})^{2}} \\ &=\frac{4(3+\sqrt{5})^{2}}{9-5} \\ &=\frac{4(3+\sqrt{5})}{4} \\ &=3+\sqrt{5} \end{aligned}$

We know that,

Formula for sum of zeros $=-\frac{b}{a}$

Thus,

The Sum of zeros $(x+y)=3+\sqrt{5}-\sqrt{5}$

The Product of zeros (x+y) = 6

Therefore, the general formula for quadratic polynomial  is  

$0=x^{2}-(\text { sum of zeros })^{2}+(\text { product of } \text { zeros })$

Therefore, the quadratic polynomial is $x^{2}-6 x+4=0$.