Question

Zeroes of the quadratic polynomial x² + x − 6 are 'α' and 'β'. Construct a quadratic polynomial whose zeroes are `1/α  "and"  1/β`.

Answer 1

The given quadratic polynomial is x² + x − 6.

To find the zeros of the given polynomial, let x² + x − 6 = 0

⇒ x² + 3x − 2x − 6 = 0

⇒ x(x + 3) − 2(x + 3) = 0

⇒ (x + 3) (x − 2) = 0

⇒ x + 3 = 0 or x − 2 = 0

⇒ x = −3, 2

According to the question, let α = − 3 and β = 2.

We have to construct a quadratic polynomial whose zeros are `1/α  "and"  1/β`

Now the quadratic equation whose zeros are `1/α  "and"  1/β` is `(x - 1/α) (x - 1/β) = 0` 

⇒ `(x + 1/3) (x - 1/2) = 0`

⇒ `x^2 - 1/6 x - 1/6 = 0`

⇒ 6x² − x − 1 = 0

Therefore, the required quadratic polynomial is 6x² − x − 1 = 0.

Answer 2

The quadratic equation is x² + x − 6

We know that for reciprocal roots, we only need to replace x by `1/x` in the given equation.

Therefore, the quadratic equation, whose roots are reciprocal roots of x² + x − 6 = 0,

`1/x^2 + 1/x - 6 = 0`

⇒ 1 + x − 6x² = 0

⇒ 6x² − x − 1 = 0

Therefore, the required quadratic polynomial is 6x² − x − 1.