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

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $- x$ .

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

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

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

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

$x + 2 = x^{2}$

Step 3

Solve for $x$ .

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

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

Factor the left side of the equation.

Factor $- 1$ out of $x + 2 - x^{2}$ .

Reorder the expression.

Move $2$ .

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

Reorder $x$ and $- x^{2}$ .

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

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + x + 2 = 0$

Factor $- 1$ out of $x$ .

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

Rewrite $2$ as $- 1 (- 2)$ .

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

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

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

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 2)$ .

$- (x^{2} - x - 2) = 0$

Factor.

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

$- 2, 1$

Write the factored form using these integers.

$- ((x - 2) (x + 1)) = 0$

Remove unnecessary parentheses.

$- (x - 2) (x + 1) = 0$

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

$x - 2 = 0$

$x + 1 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $- (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 4

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

$x = - 1$

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

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