Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 \sqrt{- x} + 2 = 0$ .

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

Step 1.2.2

Subtract $2$ from both sides of the equation.

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

Step 1.2.3

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

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

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Apply the product rule to $- x$ .

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

Evaluate the exponent.

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

Step 1.2.4.2.1.4

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

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

Step 1.2.4.2.1.5

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

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Step 1.2.4.2.1.5.1

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

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

Step 1.2.4.2.1.5.2

Apply the product rule to $2 i$ .

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

Step 1.2.4.2.1.6

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

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

Step 1.2.4.2.1.7

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

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

Step 1.2.4.2.1.8

Multiply $4$ by $- 1$ .

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

Step 1.2.4.2.1.9

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

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Step 1.2.4.2.1.9.1

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

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

Step 1.2.4.2.1.9.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.9.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.9.2.2

Rewrite the expression.

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

Step 1.2.4.2.1.10

Simplify.

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$- 4 x = 4$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $4$ by $- 4$ .

$x = - 1$

Step 1.2.6

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

$No solution$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 \sqrt{- (0)} + 2$

Step 2.2

Simplify $2 \sqrt{- (0)} + 2$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 1$ by $0$ .

$y = 2 \sqrt{0} + 2$

Step 2.2.1.2

Rewrite $0$ as $0^{2}$ .

$y = 2 \sqrt{0^{2}} + 2$

Step 2.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$y = 2 \cdot 0 + 2$

Step 2.2.1.4

Multiply $2$ by $0$ .

$y = 0 + 2$

Step 2.2.2

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2)$

Step 4