Solve for x 6=x+ square root of x

$6 = x + \sqrt{x}$

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

$x + \sqrt{x} = 6$

Subtract $x$ from both sides of the equation.

$\sqrt{x} = 6 - x$

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

${\sqrt{x}}^{2} = {(6 - x)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(6 - x)}^{2}$ .

Rewrite ${(6 - x)}^{2}$ as $(6 - x) (6 - x)$ .

$x = (6 - x) (6 - x)$

Expand $(6 - x) (6 - x)$ using the FOIL Method.

Apply the distributive property.

$x = 6 (6 - x) - x (6 - x)$

Apply the distributive property.

$x = 6 \cdot 6 + 6 (- x) - x (6 - x)$

Apply the distributive property.

$x = 6 \cdot 6 + 6 (- x) - x \cdot 6 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$x = 36 + 6 (- x) - x \cdot 6 - x (- x)$

Multiply $- 1$ by $6$ .

$x = 36 - 6 x - x \cdot 6 - x (- x)$

Multiply $6$ by $- 1$ .

$x = 36 - 6 x - 6 x - x (- x)$

Rewrite using the commutative property of multiplication.

$x = 36 - 6 x - 6 x - 1 \cdot - 1 x \cdot x$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x = 36 - 6 x - 6 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$x = 36 - 6 x - 6 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$x = 36 - 6 x - 6 x + 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$x = 36 - 6 x - 6 x + x^{2}$

Subtract $6 x$ from $- 6 x$ .

$x = 36 - 12 x + x^{2}$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$36 - 12 x + x^{2} = x$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$36 - 12 x + x^{2} - x = 0$

Subtract $x$ from $- 12 x$ .

$36 + x^{2} - 13 x = 0$

Factor $36 + x^{2} - 13 x$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $36$ and whose sum is $- 13$ .

$- 9, - 4$

Write the factored form using these integers.

$(x - 9) (x - 4) = 0$

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

$x - 9 = 0$

$x - 4 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 9) (x - 4) = 0$ true.

$x = 9, 4$

Exclude the solutions that do not make $6 = x + \sqrt{x}$ true.

$x = 4$

Solve for x 6=x+ square root of x

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