Question

Find the quadratic equation whose roots are
$(6 + \sqrt{5} )$
and

$(6 - \sqrt{5} )$
​

Answer 1

☆ Solution :-

Given,

• Roots of quadratic equation = ( 6 + √5) , (6 - √5)

We need to find ,

• Quadratic polynomial = ?

To find , quadratic polynomial firstly we need to find , sum of roots & product of roots

◉ Sum of roots = 6 + √5 + 6 - √5 = 12

◉ Product of roots = (6 + √5)(6 - √5)

Simplifying using (a + b)(a - b) = a² - b²

=> 6² - ( √5 )² = 36 - 5 = 31

Now , finding quadratic polynomial

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

Substituting known values,

⇒ x² - ( 12 )x + 31

⇒ x² - 12x + 31

Hence , quadratic polynomial is x² - 12x + 31

Answer 2

$\;\;\underline{\textbf{\textsf{ Given:-}}}$

• Roots of quadratic equation are

1.( 6 + √5)

2. (6 - √5)

$\;\;\underline{\textbf{\textsf{ To Find :-}}}$

• Quadratic polynomial

$\;\;\underline{\textbf{\textsf{ Solution :-}}}$

Let α and β be the roots of the given quadratic equation.

Hence,

α = 6 + √5

β = 6 - √5

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Now, we have to find sum of roots & product of roots of the given quadratic polynomial.

$\underline{\:\textsf{As we know that:}}$

• Sum of roots = α + β

• Product of roots = αβ

$\underline{\:\textsf{Putting the given values :-}}$

Sum of roots :-

= α + β

= 6 + √5 + 6 - √5

= 12

Product of roots :-

= αβ

=(6 + √5)(6 - √5)

= 6² - ( √5 )²

= 31

• Required quadratic polynomial

= x² - (α + β) x + αβ

= x² - ( 12 )x + 31

= x² - 12x + 31

$\;\;\underline{\textbf{\textsf{ Hence-}}}$

$\underline{\textsf{The required quadratic polynomial is \textbf{x² - 12x + 31}}}.$

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