Question

two roots of the quadratic Equation are 1-3√5 and 1+3√5 frame the equation ​?

Answer 1

Solution :-

2 roots of the equation are 1 - 3√5 and 1 + 3√5

Let α = 1 - 3√5 and β = 1 + 3√5

Sum of roots = α + β

= 1 - 3√5 + 1 + 3√5

= 2

Product of roots = αβ

= (1 - 3√5)(1 + 3√5)

= 1² - (3√5)²

= 1 - 9(5)

= 1 - 45

= - 44

General form of quadratic equation :

⇒ x² - (α + β)x + αβ = 0

[ Where α and β are roots ]

⇒ x² - 2x + ( - 44) = 0

⇒ x² - 2x - 44 = 0

Therefore x² - 2x - 44 is the required equation.

Answer 2

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Given:

Two roots of the quadratic equation are 1-3√5 & 1+3√5.

To find:

Frame the equation.

Explanation:

Let α= 1-3√5 & β= 1+3√5.

Therefore,

• Sum of the roots:

α+β = 1-3√5 + 1+3√5

α+β = 1+1

α+β = 2

&

• Product of roots:

αβ = (1-3√5)(1+3√5)

αβ = 1+3√5 -3√5 - 9×5

αβ = 1 -45

αβ = -44

Now,

The quadratic equation is, x² -(α+β)x +αβ= 0

∴ x² -2x + (-44)= 0

→ x² -2x - 44= 0