Solve for x 2x^(1/2)-x^(1/4)=0

$2 x^{\frac{1}{2}} - x^{\frac{1}{4}} = 0$

Step 1

Factor the left side of the equation.

Rewrite $x^{\frac{1}{2}}$ as ${(x^{\frac{1}{4}})}^{2}$ .

$2 {(x^{\frac{1}{4}})}^{2} - x^{\frac{1}{4}} = 0$

Let $u = x^{\frac{1}{4}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{4}}$ .

$2 u^{2} - u = 0$

Factor $u$ out of $2 u^{2} - u$ .

Factor $u$ out of $2 u^{2}$ .

$u (2 u) - u = 0$

Factor $u$ out of $- u$ .

$u (2 u) + u \cdot - 1 = 0$

Factor $u$ out of $u (2 u) + u \cdot - 1$ .

$u (2 u - 1) = 0$

Replace all occurrences of $u$ with $x^{\frac{1}{4}}$ .

$x^{\frac{1}{4}} (2 x^{\frac{1}{4}} - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{\frac{1}{4}} = 0$

$2 x^{\frac{1}{4}} - 1 = 0$

Step 3

Set $x^{\frac{1}{4}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{4}}$ equal to $0$ .

$x^{\frac{1}{4}} = 0$

Solve $x^{\frac{1}{4}} = 0$ for $x$ .

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = 0^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Rewrite the expression.

$x^{1} = 0^{4}$

Simplify.

$x = 0^{4}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 4

Set $2 x^{\frac{1}{4}} - 1$ equal to $0$ and solve for $x$ .

Set $2 x^{\frac{1}{4}} - 1$ equal to $0$ .

$2 x^{\frac{1}{4}} - 1 = 0$

Solve $2 x^{\frac{1}{4}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x^{\frac{1}{4}} = 1$

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(2 x^{\frac{1}{4}})}^{4} = 1^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(2 x^{\frac{1}{4}})}^{4}$ .

Apply the product rule to $2 x^{\frac{1}{4}}$ .

$2^{4} {(x^{\frac{1}{4}})}^{4} = 1^{4}$

Raise $2$ to the power of $4$ .

$16 {(x^{\frac{1}{4}})}^{4} = 1^{4}$

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 x^{\frac{1}{4} \cdot 4} = 1^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$16 x^{\frac{1}{4} \cdot 4} = 1^{4}$

Rewrite the expression.

$16 x^{1} = 1^{4}$

Simplify.

$16 x = 1^{4}$

Simplify the right side.

One to any power is one.

$16 x = 1$

Divide each term in $16 x = 1$ by $16$ and simplify.

Divide each term in $16 x = 1$ by $16$ .

$\frac{16 x}{16} = \frac{1}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{1}{16}$

Divide $x$ by $1$ .

$x = \frac{1}{16}$

Step 5

The final solution is all the values that make $x^{\frac{1}{4}} (2 x^{\frac{1}{4}} - 1) = 0$ true.

$x = 0, \frac{1}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{16}$

Decimal Form:

$x = 0, 0.0625$

Solve for x 2x^(1/2)-x^(1/4)=0

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