Solve for x 0 = square root of 6x-x

$0 = \sqrt{6 x} - x$

Step 1

Rewrite the equation as $\sqrt{6 x} - x = 0$ .

$\sqrt{6 x} - x = 0$

Step 2

Add $x$ to both sides of the equation.

$\sqrt{6 x} = x$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{6 x}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x}$ as ${(6 x)}^{\frac{1}{2}}$ .

${({(6 x)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(6 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 x)}^{\frac{1}{2}})}^{2}$ .

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

${(6 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(6 x)}^{1} = x^{2}$

Simplify.

$6 x = x^{2}$

Step 5

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

$6 x - x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$6 u - u^{2} = 0$

Factor $u$ out of $6 u - u^{2}$ .

Factor $u$ out of $6 u$ .

$u \cdot 6 - u^{2} = 0$

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

$u \cdot 6 + u (- u) = 0$

Factor $u$ out of $u \cdot 6 + u (- u)$ .

$u (6 - u) = 0$

Replace all occurrences of $u$ with $x$ .

$x (6 - x) = 0$

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

$x = 0$

$6 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $6 - x$ equal to $0$ and solve for $x$ .

Set $6 - x$ equal to $0$ .

$6 - x = 0$

Solve $6 - x = 0$ for $x$ .

Subtract $6$ from both sides of the equation.

$- x = - 6$

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

The final solution is all the values that make $x (6 - x) = 0$ true.

$x = 0, 6$

Solve for x 0 = square root of 6x-x

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