Question

If α , β are roots of x2 + 5x + a =0 and 2α + 5β = -1 then α is equal to _____?

Answer 1

9514 1404 393

Answer:

  -8

Step-by-step explanation:

If α and β are roots of x^2 +5x +a = 0, then we must have ...

  (x -α)(x -β) = x^2 +5x +a

  x^2 -(α+b)x +αβ = x^2 +5x +a

Then ...

  α + β = -5

Together with the other relation between the roots, we have a system of equations in α and β.

Using the above to write an expression for β, we get ...

  β = -α -5

Substituting this into the given relation, we have an equation for α:

  2α +5(-α -5) = -1

  -3α -25 = -1 . . . . . . eliminate parentheses

  -24 = 3α . . . . . . . . add 3α +1

  -8 = α

The root α is equal to -8.