Question

if alpha and beta are the roots of quadratic equation X square - 5 x + 8 = 20 then find the alpha + 1 upon alpha and beta +1 upon beta​

Answer 1

Correct question :-

If α and β are the roots of quadratic equation 3x² + kx + 8 = 0 and α/β = 2/3 then find the value of k .

Solution :-

3x² + kx + 8 = 0

Comparing with ax² + bx + c = 0 we get,

   a = 3

   b = k

   c = 8

α and β are the roots of the equation

Given : α/β = 2/3

⇒ α = (2/3) * β

Sum of roots = α + β = - b/a = - k/3

⇒ α + β = - k/3

Product of roots = αβ = c/a = 8/3

⇒ αβ = 8/3

Substituting α = (2/3) * β

⇒ (2/3) * β * β = 8/3

⇒ β² = 8/3 * 3/2

⇒ β² = 8/2

⇒ β² = 4

⇒ β = ± √4 = ± 2

⇒ β = ± 2

Substituting β = ± 2 in α = (2/3) * β

⇒ α = (2/3) * ( ± 2)

⇒ α = ± 4/3

Substituting α = ± 2 and β = ± 4/3 in α + β = - k/3

When α = 2, β = 4/3

⇒ α + β = - k/3

⇒ 2 + 4/3 = - k/3

⇒ (6 + 4)/3 = - k/3

⇒ 10 = - k

⇒ k = - 10

When α = - 2, β = - 4/3

⇒ α + β = - k/3

⇒ - 2 - 4/3 = - k/3

⇒ (-6 - 4)/3 = - k/3

⇒ - 10 = - k

⇒ k = 10

Therefore the value of k is 10 or - 10.

Answer 2

Step-by-step explanation:

Please see the attachment