Solve for x x- square root of x-2=0

$x - \sqrt{x} - 2 = 0$

Step 1

Move all terms not containing $- \sqrt{x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{x} - 2 = - x$

Add $2$ to both sides of the equation.

$- \sqrt{x} = - x + 2$

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

Raise $- 1$ to the power of $2$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x = (- x + 2) (- x + 2)$

Expand $(- x + 2) (- x + 2)$ using the FOIL Method.

Apply the distributive property.

$x = - x (- x + 2) + 2 (- x + 2)$

Apply the distributive property.

$x = - x (- x) - x \cdot 2 + 2 (- x + 2)$

Apply the distributive property.

$x = - x (- x) - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x = - 1 \cdot - 1 x \cdot x - x \cdot 2 + 2 (- x) + 2 \cdot 2$

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

Move $x$ .

$x = - 1 \cdot - 1 (x \cdot x) - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

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

$x = x^{2} - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $2$ by $- 1$ .

$x = x^{2} - 2 x + 2 (- x) + 2 \cdot 2$

Multiply $- 1$ by $2$ .

$x = x^{2} - 2 x - 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$x = x^{2} - 2 x - 2 x + 4$

Subtract $2 x$ from $- 2 x$ .

$x = x^{2} - 4 x + 4$

Step 4

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.

$x^{2} - 4 x + 4 = x$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 4 x + 4 - x = 0$

Subtract $x$ from $- 4 x$ .

$x^{2} - 5 x + 4 = 0$

Factor $x^{2} - 5 x + 4$ 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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$(x - 4) (x - 1) = 0$

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

$x - 4 = 0$

$x - 1 = 0$

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.

$x = 4$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 4) (x - 1) = 0$ true.

$x = 4, 1$

Step 5

Exclude the solutions that do not make $x - \sqrt{x} - 2 = 0$ true.

$x = 4$

Solve for x x- square root of x-2=0

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