Find All Complex Solutions square root of x+6=x

$\sqrt{x} + 6 = x$

Step 1

Subtract $x$ from both sides of the equation.

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

Step 2

Factor the left side of the equation.

Factor $- 1$ out of $\sqrt{x} + 6 - x$ .

Reorder the expression.

Move $6$ .

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

Reorder $\sqrt{x}$ and $- x$ .

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

Factor $- 1$ out of $- x$ .

$- (x) + \sqrt{x} + 6 = 0$

Factor $- 1$ out of $\sqrt{x}$ .

$- (x) - 1 (- \sqrt{x}) + 6 = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$- (x) - 1 (- \sqrt{x}) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (x) - 1 (- \sqrt{x})$ .

$- (x - \sqrt{x}) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (x - \sqrt{x}) - 1 (- 6)$ .

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

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

$- (x - x^{\frac{1}{2}} - 6) = 0$

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

$- ({(x^{\frac{1}{2}})}^{2} - x^{\frac{1}{2}} - 6) = 0$

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

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

Factor $u^{2} - u - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$- ((u - 3) (u + 2)) = 0$

Factor.

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$- ((x^{\frac{1}{2}} - 3) (x^{\frac{1}{2}} + 2)) = 0$

Remove unnecessary parentheses.

$- (x^{\frac{1}{2}} - 3) (x^{\frac{1}{2}} + 2) = 0$

Step 3

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

$x^{\frac{1}{2}} - 3 = 0$

$x^{\frac{1}{2}} + 2 = 0$

Step 4

Set $x^{\frac{1}{2}} - 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 3$ equal to $0$ .

$x^{\frac{1}{2}} - 3 = 0$

Solve $x^{\frac{1}{2}} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{\frac{1}{2}} = 3$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the exponent.

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

$x^{1} = 3^{2}$

Simplify.

$x = 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$x = 9$

Step 5

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

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

$x^{\frac{1}{2}} + 2 = 0$

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

Subtract $2$ from both sides of the equation.

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

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x = 4$

Step 6

The final solution is all the values that make $- (x^{\frac{1}{2}} - 3) (x^{\frac{1}{2}} + 2) = 0$ true.

$x = 9, 4$

Find All Complex Solutions square root of x+6=x

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