Question

If the equations x² − ax + b = 0 and x² − ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f)

Answer 1

The given quadratic equations are

x² – ax + b = 0  .....(1)

x² – ex + f = 0   .....(2)

Let α be the common root of the given quadratic equations (1) and (2)

Let α, β be the roots of x² – ax + b = 0

Sum of the roots α + β = `−(− "a"/1)`

α + β = a   ......(3)

Product of the roots αβ = `"b"/1`

αβ = b   ......(4)

Given that the second equation has equal roots.

∴ The roots of the second equation are a, a

Sum of the roots α + α = `−(−"e"/1)`

2α = e   ......(5)

Product of the roots α.α = `"f"/1`

α² = f   ......(6)

ae = (α + β)2α (Multiplying equations (3) and (5))

ae = 2α² + 2αβ

ae= 2(f) + 2b

From equations (4) and (6)

ae= 2(f + b)

Hence proved.