How do you solve #x^3 -|x|=0#?

1 Answer
MattyMatty
Feb 11, 2018

#x = 0 or 1#

Explanation:

#"Apply the definition of |x| : "#
#= {( x ", " x >= 0), (-x", " x<=0) :}#

#"Now take first the case "x >= 0 " : "#
#"Then we have"#
#x^3 - x = 0#
#=> x(x^2 - 1) = 0#
#=> x(x-1)(x+1) = 0#
#=> x = cancel(-1), 0, or 1#
#"We eliminate x= -1 as we made the assumption "x>=0"."#

#"Then take the second case "x <= 0" : "#
#"Then we have"#
#x^3 + x = 0#
#=> x(x^2 + 1) = 0#
#=> x = 0 " (x² + 1 > 0 and has no real solutions)"#

#"So we have only x=0, or 1 as real solution of the equation."#

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