Question

The quadratic equation `3x^2 -9x +b =0` has roots `alpha` and `alpha+2`, find `b`?

Answer 1

`b=15/4`

Explanation:

Suppose the roots of the general quadratic equation:

`ax^2+bx+c = 0`

are `alpha` and `beta` , then using the root properties we have:

`"sum of roots" \ \ \ \ \ \= alpha+beta = -b/a`
`"product of roots" = alpha beta \ \ \ \ = c /a`

So for the given quadratic with roots `alpha` and `beta`:

`3x^2 -9x +b =0`

we know that:

`alpha+beta = -(-9)/3=3 \ \ \` ; and `\ \ \ alpha beta = b/3`

But we also know that `beta=alpha + 2`

Hence,

`alpha+beta= = alpha+(alpha+2) = 2alpha+2 => 2alpha+2=3`
`:. alpha = 1/2`

and:

`alpha beta= = alpha(alpha+2) = 1/2(1/2+2) = 5/4`
`:. 5/4 = b/3 => b=15/4`