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