Which of the following numbers does b have to be divisible by for the equation ax^2 + bx + c = 0 to have equal roots: 3, 4, 5, 6?
The roots of a quadratic equation ax^2 + bx + c = 0 can be expressed as [-b + sqrt (b^2 – 4ac)]/2a and [-b - sqrt (b^2 – 4ac)]/2a.
If the two roots have to be equal b^2 – 4ac should be equal to 0.
Equating b^2 – 4ac to 0
=> b^2 = 4ac
=> b has to be a multiple of 4.
Therefore for a quadratic equation ax^2 + bx + c = 0 to have equal roots the coefficient b has to be a multiple of 4, irrespective of the value of a and c.
We'll use Viete's relations, to express the condition for the equation to have 2 equal roots:
x1 = x2
x1 + x2 = -b/a
But x1 = x2 = x
2x = -b/a
We'll divide by 2:
x = -b/2a (1)
x*x = c/a
b^2/4a^2 = c/a
b^2 = 4a^2*c/a
We'll simplify and we'll get:
b^2 = 4ac (2)
It is obvious that from the series of the given numbers and from Viete's relations (1) and (2), only 4 keeps the terms of condition.