Find the quadratic equation whose roots are $9 - \sqrt{5}, 9 + \sqrt{5}$ .

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x2−18x−56=0

x2+18x+56=0

4x2−72x−56=0

x2−18x+76=0

answer is D.

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Detailed Solution

We are given to find the quadratic equation whose roots are

9 - \sqrt{5}, 9 + \sqrt{5}

.
Let

9 - \sqrt{5}

be ‘a’ and

9 + \sqrt{5}

be ‘b’.
When the roots of a quadratic equation are ‘a’ and ‘b’, then the equation will be x2−(a+b)x+ab=0
Where a+b is the sum of the roots and ab is the product of the roots of the quadratic equation
a+b =

9 - \sqrt{5} + 9 + \sqrt{5}

= 9+9 = 18
Canceling the similar terms with different signs, we get a+b = 9+9 = 18
ab=(

9 - \sqrt{5}) (9 + \sqrt{5}

)
The RHS of the above equation is in the form (x−y)(x+y) which is equal to x2−y2
Therefore,

ab = 9^{2} - {(\sqrt{5})}^{2}

⇒ab=81−5=76
On substituting the values of obtained a+b and ab in x2−(a+b)x+ab=0 , we get
⇒x2−(18)x+76=0
Therefore the quadratic equation whose roots are