If the difference between the roots of ax^2 + bx + c = 0 is 1

• Quadratic Equations

If the difference between the roots of ax² + bx + c = 0 is 1, then which one of the following is correct?

1. b² = a(a + 4c)
2. a² = b(b + 4c)
3. a² = c(a + 4c)
4. b² = a(b + 4c)

Solution

Let α and β be the roots of the equation.

Sum = α + β = -b/a

Product = αβ = c/a

α - β = 1

(α - β)² = α² + β² - 2αβ

(α - β)² = (α + β)² - 4αβ

1 = (-b/a)² - 4c/a

1 + 4c/a = b²/a²

a² + 4ac = b²

b² = a(a + 4c)

The correct option is A.