Find All Complex Solutions 4+ square root of x-2=x

$4 + \sqrt{x - 2} = x$

Subtract $4$ from both sides of the equation.

$\sqrt{x - 2} = x - 4$

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

${\sqrt{x - 2}}^{2} = {(x - 4)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x - 2 = {(x - 4)}^{2}$

Simplify the right side.

Simplify ${(x - 4)}^{2}$ .

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$x - 2 = (x - 4) (x - 4)$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$x - 2 = x (x - 4) - 4 (x - 4)$

Apply the distributive property.

$x - 2 = x \cdot x + x \cdot - 4 - 4 (x - 4)$

Apply the distributive property.

$x - 2 = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

$x - 2 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$x - 2 = x^{2} - 4 x - 4 x + 16$

Subtract $4 x$ from $- 4 x$ .

$x - 2 = x^{2} - 8 x + 16$

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.

$x^{2} - 8 x + 16 = x - 2$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 8 x + 16 - x = - 2$

Subtract $x$ from $- 8 x$ .

$x^{2} - 9 x + 16 = - 2$

Add $2$ to both sides of the equation.

$x^{2} - 9 x + 16 + 2 = 0$

Add $16$ and $2$ .

$x^{2} - 9 x + 18 = 0$

Factor $x^{2} - 9 x + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$(x - 6) (x - 3) = 0$

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

$x - 6 = 0$

$x - 3 = 0$

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x - 6) (x - 3) = 0$ true.

$x = 6, 3$

Exclude the solutions that do not make $4 + \sqrt{x - 2} = x$ true.

$x = 6$

Find All Complex Solutions 4+ square root of x-2=x

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