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?  

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.  

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