Question

If alpha and beta are the roots of the equation x^2+px+q=0,then find the value of alpha/beta+beta/alpha.

Answer 1

AnswEr

The value of α/β + β/α (p² - 2q)/q

Solution

Given ,

The quadratic equation is

x² + px + q = 0

The roots of the equation are α and β

To find →

The value of

(α/β )+ (β/α)

We know that

Sum of the roots = -co-efficient of x/co-efficient of x²

⇒α + β = -p/1

⇒ α + β = -p

⇒(α + β)² = p²

⇒α² + β² + 2αβ= p² ________(1)

Again

Product of the roots = constant term/co-efficient of x²

⇒αβ = q/1

⇒ αβ = q ___________(2)

Using the value of (2) in (1) we have

⇒ α² + β² + 2q = p²

⇒α² + β² = p² - 2q __________(3)

Now diving (3) by (2) we get

α²/αβ + β²/αβ = (p² - 2q)/q

⇒ α/β + β/α = (p² - 2q)/q

Thus the value of α/β + β/α is

(p² - 2q)/q