Q:

What are the solutions to the equation 4x 3 - 5x = |4x|? List your answers in increasing order.The solutions are x = , and

Accepted Solution

A:
Answer:-1/2 , 0 , 3/2Step-by-step explanation:Given equation is:[tex]4x^3-5x = |4x|[/tex]We know that [tex]|x|=a\\The\ solution\ will\ be:\\x=a\ and\ x=-a\\[/tex]So, from given equation,we will get two solutions:[tex]4x^3-5x = 4x\\4x^3-5x-4x=0\\4x^3-9x=0\\x(4x^2-9) = 0\\x = 0\\and\\4x^2-9 = 0\\4x^2=9\\x^2 = \frac{9}{4} \\\sqrt{x^2}=\sqrt{\frac{9}{4} }\\[/tex]x= ±√3/2 , 0and[tex]4x^3-5x = -4x\\4x^3-5x+4x=0\\4x^3-x=0\\x(4x^2-1) = 0\\x = 0\\and\\4x^2-1 = 0\\4x^2=1\\x^2 = \frac{1}{4} \\\sqrt{x^2}=\sqrt{\frac{1}{4} }[/tex]x= ±1/2 , 0We can check that 1/2 and -3/2 do not satisfy the given equation.[tex]4x^3-5x = |4x|\\Put\ x=1/2\\4(\frac{1}{2})^3 - 5(\frac{1}{2}) = |4 * \frac{1}{2}|\\   4 * (\frac{1}{8)} - \frac{5}{2} = |2|\\ -2 = 2\\Put\ x=-\frac{3}{2} \\4(\frac{-3}{2})^3 - 5(\frac{-3}{2}) = |4 * \frac{-3}{2}|\\-6 = 6\\[/tex]So, 1/2 and -3/2 will not be the part of the solution ..So, the solutions in increasing order are:-1/2 , 0 , 3/2 ..