In the equation \(3x^2−12x+b=0\) with equal roots, if the positions of b and 12 are interchanged, then the roots of the new equation are ____.

Given:

Original equation: 3x² − 12x + b = 0 with equal roots

Concept Used:

For equal roots, discriminant (Δ) = 0

Discriminant formula: Δ = b² − 4ac

Calculation:

For the original equation, a = 3, b = -12, c = b (unknown)

Δ = (-12)² − 4 × 3 × b = 144 − 12b = 0

⇒ 144 = 12b

⇒ b = 12

New equation:

If the positions of b and 12 are interchanged, the new equation becomes:

3x² − 12x + 12 = 0

Discriminant of the new equation:

Δ = (-12)² − 4 × 3 × 12 = 144 − 144 = 0

Since the discriminant is 0, the new equation also has equal roots.

Roots of the new equation:

Using the quadratic formula: x = (-b ± √Δ) / 2a

For 3x² − 12x + 12 = 0, a = 3, b = -12, Δ = 0

⇒ x = (-(-12) ± √0) / (2 × 3) = 12 / 6 = 2

∴ The roots of the new equation are x = 2, 2 (equal roots)