Precalculus Examples

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

Step 1

Set $\sqrt{4 x - x^{2}}$ equal to $0$ .

$\sqrt{4 x - x^{2}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

${\sqrt{4 x - x^{2}}}^{2} = 0^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - x^{2}}$ as ${(4 x - x^{2})}^{\frac{1}{2}}$ .

${({(4 x - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

Multiply the exponents in ${({(4 x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(4 x - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(4 x - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(4 x - x^{2})}^{1} = 0^{2}$

Step 2.2.2.1.2

Simplify.

$4 x - x^{2} = 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 x - x^{2} = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Factor the left side of the equation.

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Step 2.3.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u - u^{2} = 0$

Step 2.3.1.2

Factor $u$ out of $4 u - u^{2}$ .

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Step 2.3.1.2.1

Factor $u$ out of $4 u$ .

$u \cdot 4 - u^{2} = 0$

Step 2.3.1.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 4 + u (- u) = 0$

Step 2.3.1.2.3

Factor $u$ out of $u \cdot 4 + u (- u)$ .

$u (4 - u) = 0$

Step 2.3.1.3

Replace all occurrences of $u$ with $x$ .

$x (4 - x) = 0$

Step 2.3.2

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

$x = 0$

$4 - x = 0$

Step 2.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.4

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 2.3.4.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 2.3.4.2

Solve $4 - x = 0$ for $x$ .

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Step 2.3.4.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 2.3.4.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 2.3.4.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 2.3.4.2.2.2

Simplify the left side.

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Step 2.3.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 2.3.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 2.3.4.2.2.3

Simplify the right side.

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Step 2.3.4.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 2.3.5

The final solution is all the values that make $x (4 - x) = 0$ true.

$x = 0, 4$

Step 3