Find the Roots (Zeros) f(x)=2 square root of -x+2

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

Set $2 \sqrt{- x} + 2$ equal to $0$ .

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

Solve for $x$ .

Subtract $2$ from both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $- x$ .

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

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

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

Evaluate the exponent.

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

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

Apply the product rule to $2 i$ .

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

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

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

Rewrite $i^{2}$ as $- 1$ .

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

Multiply $4$ by $- 1$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$- 4 x = 4$

Divide each term in $- 4 x = 4$ by $- 4$ and simplify.

Divide each term in $- 4 x = 4$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{4}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{4}{- 4}$

Divide $x$ by $1$ .

$x = \frac{4}{- 4}$

Simplify the right side.

Divide $4$ by $- 4$ .

$x = - 1$

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

$No solution$

Find the Roots (Zeros) f(x)=2 square root of -x+2

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