Question

Find the quadratic polinomail whose one zero is2+√5

Answer 1

Question :-

→ Find the quadratic polinomail whose one zero is2+√5 .

Answer :-

$\implies \: \red{ \boxed{\bf{{x}^{2} - 4x - 1 = 0}}} \:$

To find :-

Find quadratic polynomial.

Formula used :-

• General quadratic equation is ↓

$\: \small \: \bf{{x}^{2} - (sum \: of \: zeros)x + product \: of \: zeros \: }$

$\star \: \boxed{ \bf{(x + y)(x - y) = {x}^{2} - {y}^{2} }}$

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Explanation :-

According to the question,

Given that ,

One zero of the required polynomial is

→ 2+√5

This is an irrational number ,

We know that ,if zero of a quadratic polynomial are irrational then the zeros of polynomial is always in pair of irrational zeros,

Hence , Pair of irrational zeroes is

( p + √q) and ( p - √q)

Therefore ,

Another zero is ( 2 - √5)

$\star \: \small \: sum \: of \: zeros \: = 2 + \sqrt{5} + 2 - \sqrt{5} \\ \implies \: 4 \\ \\ \star \: \small \: product \: of \: zeros \: = (2 + \sqrt{5} )(2 - \sqrt{5} ) \\ \\ \implies \: {2}^{2} - { (\sqrt{5}) }^{2} = 4 - 5 = 1$

Therefore,

The required quadratic polynomial is ↓

$\implies \: \red{ \boxed{\bf{{x}^{2} - 4x - 1 = 0}}}$

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• For verification

We can put anyone zero in quadratic equation then we will get zero.

If we put x = (2+√5)

→ (2+√5)^{2} - 4(2+√5) -1 = 0

→ 4 + 5 + 4√5 - 8 -4√5 -1 = 0

→ 9-9 = 0

→ 0 = 0

Answer veriefied .

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Answer 2

It is given that the two roots of the polynomial are 2 and −5.

Let 

Now, sum of the zeroes,  = 2 + (−5) = −3

Product of the zeroes,  = 2−5 = −10

∴ Required polynomial = 

=x2—(−3)x+(−10)=x2+3x−10

Hope it's help full for u