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

1 Answer
Apr 1, 2017

Answer:

# 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#