For what value of b are the roots of the equation 3x^2 - bx + 15 = 0 equal.

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mitch1 | (Level 1) Adjunct Educator

Posted on

Solution:

Let us look for the value of b using factoring.

We know that we will have to make it in the form  (3x  - a )(x- c )=0, since the leading coefficient is 3x^2 and both minus because the constant is positive and the middle term is negative. Multiplying two negatives gives a positive and adding adding two negatives is negative.

We have to find the value of b that will give equal roots. So we solve for the roots first in the equation (3x  - a )(x- c )=0

Using the zero factor theorem we have,

3x - a = 0     or   x - c = 0 

x = a/3         or    x = c 

To make the roots equal we will have to do a/3 = c .

That means we have to plug-in a/3  for c  in (3x  - a )(x- c )=0  .

That makes it (3x  - a )(x- a/3 )=0

Expand:  3x^2-2ax + a^2/3 = 0

We compare this to the original equation 3x^2 -bx + 15= 0

So, a^2/3 = 15  and  2a =  b .

Solve for the value of a in a^2/3 = 15

a = +3 sqrt of 5

That means we have b = 2(3sqrt of 5)  = 6sqrt of 5 .

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The roots of a quadratic equation ax^2 + bx + c = 0 are given by `(-b+-sqrt(b^2 - 4ac))/(2a)` . If the roots of a quadratic equation are equal, `b^2 - 4ac = 0` .

For the equation given in the problem, 3x^2 - bx + 15 = 0, a = 3 and c = 15. 4*a*c = 3*4*15 = 180

b^2 = 180

=> `b = +-sqrt180`

The value of b for which the roots of the equation 3x^2 - bx + 15 = 0 are equal are `+-sqrt180`

[Check: The roots of the equations `3x^2 + sqrt180*x + 15 = 0` are `-sqrt 5` and the roots of `3x^2 + sqrt180*x + 15 = 0` are `sqrt 5` ]

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