#alpha, beta# are real numbers such that #alpha^3-3alpha^2+5 alpha - 17=0# and #beta^3-3beta^2+5beta+11 = 0#. What is the value of #alpha+beta# ?

Answer 1

#alpha+beta=2#

First use Tschirnhaus transformations to simplify the cubics:

#0 = alpha^3-3alpha^2+5alpha-17 = gamma^3+2gamma-14#
#0 = beta^3-3beta^2+5beta+11 = delta^3+2delta+14#
where #gamma = alpha-1# and #delta = beta-1#
The discriminant #Delta# of a cubic polynomial in the form #ax^3+bx^2+cx+d# is given by the formula:
#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#
We can check the discriminant of these cubics, finding that they are both negative, so they have #1# Real root and #2# Complex roots each.
By Descartes' rule of signs, the Real roots result in #gamma > 0# and #delta < 0#. We use this later.

Adding these two equations, we get:

#0 = gamma^3+delta^3+2gamma+2delta#
#color(white)(0) = (gamma+delta)(gamma^2-gamma delta+delta^2)+2(gamma+delta)#
#color(white)(0) = (gamma+delta)(gamma^2-gamma delta+delta^2+2)#
So either #gamma+delta = 0# or #gamma^2-gamma delta+delta^2+2 = 0#
Note however that #gamma*delta < 0#, hence #gamma^2-gamma delta+delta^2 + 2 > 0# for any Real values of #gamma# and #delta#.
So the only Real solution gives us #gamma+delta = 0# and hence #alpha+beta = 2#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The value of alpha + beta is 6.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7