   Chapter 8.5, Problem 26E

Chapter
Section
Textbook Problem

In Exercises 23-28, characterize the solutions to the following equations by evaluating the discriminant D 2 . x 3 −     124 x −     240 =     0

To determine

The solutions to the equation x3124x240=0 by evaluating the discriminant D2.

Explanation

Formula used:

1) Theorem: Change of variable in the cubic

The change of variable x=ya3 in x3+ax2+bx+c=0 yields the equation y3+py+q=0, where p=ba23,q=cab3+2a327.

2) Theorem: Discriminant of a cubic polynomial

The discriminant of f(y)=y3+py+q is D2=27q24p3.

3) Theorem: Real solutions of a cubic equation

The equation y3+py+q=0 has exactly three real solutions, if and only if D20; that is, if and only if 27q24p30.

Explanation:

Consider the given equation x3124x240=0

By using theorem, the change of variable x=ya3 in x3+ax2+bx+c=0 yields the equation y3+py+q=0, where p=ba23,q=cab3+2a327.

By comparing x3124x240=0 with x3+ax2+bx+c=0 gives a=0,b=124 and c=240

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