Question

If ‘α’ and ‘β’ are roots of a quadratic equation such that, 1/α + 1/β = 8/15 and α² – β²= 16, then find that quadratic equation.
1. x² + 8x + 15 = 0
2. x² – 8x – 15 = 0
3. x² + 8x – 15 = 0
4. x² – 8x + 15 = 0

Answer 1

Correct Answer - Option 4 : x² – 8x + 15 = 0

GIVEN:

1/α + 1/β = 8/15

α² – β² = 16

CONCEPT:

Roots of equation

CALCULATION:

1/α + 1/β = 8/15

⇒ (α + β)/αβ = 8/15

Let

(α + β) = 8k

αβ = 15k

α² – β² = 16

⇒ (α – β) (α + β) = 16

⇒ (α – β) 8k = 16

⇒ (α – β) = 2/k

Now,

α = (4k + 1/k)

β = (4k – 1/k)

αβ = 15k

⇒ (4k + 1/k) (4k – 1/k) = 15k

After putting k = 1

5 × 3 = 15

⇒ 15 = 15

Hence,

Required equation = x² – (α + β)x + αβ = 0

⇒ x² – 8kx + 15k = 0

⇒ x² – 8x + 15 = 0