What is the condition for one root of the quadratic equation ax2 + bx + c = 0…

CONCEPT

For a quadratic equation ax2 + bx + c = 0 with a ≠ 0 and roots α and β, Vieta’s formulas give α + β = −b/a and αβ = c/a. A prescribed ratio between the roots can therefore be imposed on these two identities to eliminate the root variable.

APPLICATION

1. Let the two roots be r and 2r; their order does not matter.

2. Using the sum of roots, 3r = −b/a, so b = −3ar.

3. Using the product of roots, 2r2 = c/a, so c = 2ar2.

4. Squaring b = −3ar gives b2 = 9a2r2. Hence 2b2 = 18a2r2, while 9ac = 9a(2ar2) = 18a2r2.

CROSS-CHECK

1. For sufficiency, let the roots be α and β. Vieta gives b = −a(α + β) and c = aαβ.

2. Substituting these into 2b2 = 9ac and cancelling a2 gives 2(α + β)2 = 9αβ.

3. Rearranging gives 2α2 − 5αβ + 2β2 = 0, which factors as (2α − β)(α − 2β) = 0.

4. Thus β = 2α or α = 2β, so either root is twice the other. This proves the condition is sufficient as well as necessary.

Therefore, the required condition is 2b2 = 9ac.