What is the answer to x^{1/3}+2x^{2/3}-x=0 ?

The answer to x^{1/3}+2x^{2/3}-x=0 is x=0,x=7+5*2^{1/2},x=7-5*2^{1/2}

Solve for x: x^{1/3}+2x^{2/3}-x=0

The solution is x=0,x=7+5*2^{1/2},x=7-5*2^{1/2}

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x^{\frac{1}{3}} + 2 x^{\frac{2}{3}} - x = 0

x = 0, x = 7 + 5 \cdot 2^{\frac{1}{2}}, x = 7 - 5 \cdot 2^{\frac{1}{2}}

Decimal Notation

x = 0, x = 14.07106 \dots, x = - 0.07106 \dots

Solution steps

x^{\frac{1}{3}} + 2 x^{\frac{2}{3}} - x = 0

Use the following exponent property

: a^{\frac{n}{m}} = (a^{\frac{1}{m}})^{n}

x^{\frac{2}{3}} = (x^{\frac{1}{3}})^{2}

x^{\frac{1}{3}} + 2 (x^{\frac{1}{3}})^{2} - (x^{\frac{1}{3}})^{3} = 0

Rewrite the equation with

x^{\frac{1}{3}} = u

u + 2 u^{2} - u^{3} = 0

Solve

u + 2 u^{2} - u^{3} = 0 : u = 0, u = 1 + 2^{\frac{1}{2}}, u = 1 - 2^{\frac{1}{2}}

u = 0, u = 1 + 2^{\frac{1}{2}}, u = 1 - 2^{\frac{1}{2}}

Substitute back

u = x^{\frac{1}{3}},

solve for

x

Solve

x^{\frac{1}{3}} = 0 : x = 0

Solve

x^{\frac{1}{3}} = 1 + 2^{\frac{1}{2}} : x = 7 + 5 \cdot 2^{\frac{1}{2}}

Solve

x^{\frac{1}{3}} = 1 - 2^{\frac{1}{2}} : x = 7 - 5 \cdot 2^{\frac{1}{2}}

x = 0, x = 7 + 5 \cdot 2^{\frac{1}{2}}, x = 7 - 5 \cdot 2^{\frac{1}{2}}

Verify Solutions:

x = 0

True

, x = 7 + 5 \cdot 2^{\frac{1}{2}}

True

, x = 7 - 5 \cdot 2^{\frac{1}{2}}

True

The solutions are

x = 0, x = 7 + 5 \cdot 2^{\frac{1}{2}}, x = 7 - 5 \cdot 2^{\frac{1}{2}}

3 x - 4 = 12

\frac{( 2 x + 1 )}{( 4 x ^{2} - 5 )} = - 1

2 x - 1 = 3

2 x - 2 = \frac{( x + 7 )}{4}

x, \frac{x}{y} + \frac{y}{x} = \frac{10}{3}

What is the answer to x^{1/3}+2x^{2/3}-x=0 ?
The answer to x^{1/3}+2x^{2/3}-x=0 is x=0,x=7+5*2^{1/2},x=7-5*2^{1/2}
Solve for x: x^{1/3}+2x^{2/3}-x=0
The solution is x=0,x=7+5*2^{1/2},x=7-5*2^{1/2}