Respuesta :
-1/2, 0, 3/2
this should be the answer, hopefully helped
this should be the answer, hopefully helped
Answer: The solutions of the given equation in increasing order are
[tex]x=-\dfrac{1}{2},~0,~\dfrac{3}{2}.[/tex]
Step-by-step explanation: The given equation is
[tex]4x^3-5x=|4x|~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to solve the above equation and to list the solutions in increasing order.
We know that
[tex]|x|=a~~~~~~\Rightarrow a=x~~\textup{or}~~a=-x.[/tex]
So, from equation (i), we get
[tex]4x^3-5x=4x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)\\\\\textup{or}\\\\4x^3-5x=-4x~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iv)[/tex]
Solving equation (iii), we get
[tex]4x^3-5x=4x\\\\\Rightarrow 4x^3-9x=0\\\\\Rightarrow x(4x^2-9)=0\\\\\Rightarrow x=0,~~4x^2-9=0~~\Rightarrow x^2=\dfrac{9}{4}~~\Rightarrow x=\pm\dfrac{3}{2}.[/tex]
So, solutions of equation (i) are [tex]x=0,~\dfrac{3}{2},~-\dfrac{3}{2}.[/tex]
And, solving equation (iv), we get
[tex]4x^3-5x=-4x\\\\\Rightarrow 4x^3-x=0\\\\\Rightarrow x(4x^2-1)=0\\\\\Rightarrow x=0,~~~4x^2-1=0~~\Rightarrow x^2=\dfrac{1}{4}~~~\Rightarrow x=\pm\dfrac{1}{2}.[/tex]
So, solutions of equation (ii) are [tex]x=0,~\dfrac{1}{2},~-\dfrac{1}{2}.[/tex]
We can see that [tex]x=\dfrac{1}{2}~~\textup{and}~~x=-\dfrac{3}{2}[/tex] does not satisfy equation (i).
For [tex]x=\dfrac{1}{2},[/tex] we have
[tex]L.H.S.=4\times\dfrac{1}{8}-5\times\dfrac{1}{2}=-2,\\\\R.H.S.=|4\times\dfrac{1}{2}|=2\neq L.H.S.[/tex]
Similarly, for [tex]x=-\dfrac{3}{2},[/tex] we have
[tex]L.H.S.=4\times -\dfrac{27}{8}+5\times\dfrac{3}{2}=-6,\\\\R.H.S.=|4\times -\dfrac{3}{2}|=6\neq L.H.S.[/tex]
Thus, the solutions of the given equation in increasing order are
[tex]x=-\dfrac{1}{2},~0,~\dfrac{3}{2}.[/tex]
