Question

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

The solutions are x =
,
and

Answer 1

`-(1)/(2),0,(3)/(2)`

Explanation:

We are given that an equation

`4x^3-5x=\mid x\mid`

We have to find the solution of given equation and arrange the solution in increasing order.

`4x^3-5x=4x` when x >0

and
`4x^3-5x=-4x` when x < 0

because
`\mid x\mid =x when x > 0`

=-x when x < 0

`4x^3-5x-4x=0`

`4x^3-9x=0`

`x(4x^2-9)=0`

`x(2x+3)(2x-3)=0`

Using identity
`a^2-b^2=(a+b)(a-b)`

`x=0,2x+3=0,2x-3=0`

`2x=3\implies x=(3)/(2)=1.5`

`2x=-3 \implies x=-(3)/(2)=-1.5`

`4x^3-5x=-4x=0`

`4x^3-5x+4x=0`

`4x^3-x=0`

`x(4x^2-1)=0`

`x(2x+1)(2x-1)=0`

`x=0,2x+1=0`

`2x-1=0`

`2x-1=0`

`2x=1 \ilmplies x=(1)/(2)=0.5`

`2x+1=0`

`2x=-1 \implies x=-(1)/(2)=-0.5`

When we substitute x=
`(1)/(2)`

`4((1)/(2))^3-(5)/(2)=(1)/(2)-(5)/(2)=(1-5)/(2)=-2`

`\mid 4((1)/(2))\mid=2`

`-2\\eq 2`

Hence,
`(1)/(2)` is a not solution of given equation.

When substitute
`x=(-3)/(2)`

`4((-3)/(2))^3+(15)/(2)=(-27)/(2)+(15)/(2)=(-27+15)/(2)=-6`

`\mid 4(-(3)/(2)\mid=6`

`-6\\eq 6`

Hence,
`(-3)/(2)` is not a solution of given equation.

Substitute x=
`-(1)/(2)` in the given equation

`4(-(1)/(2))^3+(5)/(2)=-(1)/(2)+(5)/(2)=2`

`\mid 4(-(1)/(2))\mid=2`

`2=2`

Hence,
`-(1)/(2)` is a solution of given equation.

Substitute
`x=(3)/(2)` in the given equation

`4((3)/(2))^3-(15)/(2)=(27-15)/(2)=6`

`\mid 4((3)/(2))\mid =6`

`6=6`

Hence,
`(3)/(2)` is a solution of given equation.

Answer:
`-(1)/(2),0,(3)/(2)`

Answer 2

-1/2 , 0 , 3/2

Explanation:

Given equation is:

`4x^3-5x = |4x|`

We know that
`|x|=a\\The\ solution\ will\ be:\\x=a\ and\ x=-a\\`

So, from given equation,we will get two solutions:

`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 = (9)/(4) \\√(x^2)=\sqrt{(9)/(4) }\\`

x= ±√3/2 , 0

and

`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 = (1)/(4) \\√(x^2)=\sqrt{(1)/(4) }`

x= ±1/2 , 0

We can check that 1/2 and -3/2 do not satisfy the given equation.

`4x^3-5x = |4x|\\Put\ x=1/2\\4((1)/(2))^3 - 5((1)/(2)) = |4 * (1)/(2)|\\   4 * ((1)/(8)) - (5)/(2) = |2|\\ -2 = 2\\Put\ x=-(3)/(2) \\4((-3)/(2))^3 - 5((-3)/(2)) = |4 * (-3)/(2)|\\-6 = 6\\`

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 ..