Algebra Examples

f (x) = 4 x \sqrt{3 - x}

$f (x) = 4 x \sqrt{3 - x}$

Step 1

Set

4 x \sqrt{3 - x}

$4 x \sqrt{3 - x}$ equal to

0

$0$ .

4 x \sqrt{3 - x} = 0

$4 x \sqrt{3 - x} = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

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

{(4 x \sqrt{3 - x})}^{2} = 0^{2}

${(4 x \sqrt{3 - x})}^{2} = 0^{2}$

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}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3 - x}

$\sqrt{3 - x}$ as

{(3 - x)}^{\frac{1}{2}}

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

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

${(4 x {(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify

{(4 x {(3 - x)}^{\frac{1}{2}})}^{2}

${(4 x {(3 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

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

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Step 2.2.2.1.1.1

Apply the product rule to

4 x {(3 - x)}^{\frac{1}{2}}

$4 x {(3 - x)}^{\frac{1}{2}}$ .

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

${(4 x)}^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2.1.1.2

Apply the product rule to

4 x

$4 x$ .

4^{2} x^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}

$4^{2} x^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

4^{2} x^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}

$4^{2} x^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2.1.2

Raise

4

$4$ to the power of

2

$2$ .

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

$16 x^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2.1.3

Multiply the exponents in

{({(3 - x)}^{\frac{1}{2}})}^{2}

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

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Step 2.2.2.1.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

16 x^{2} {(3 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}

$16 x^{2} {(3 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.3.2

Cancel the common factor of

2

$2$ .

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Step 2.2.2.1.3.2.1

Cancel the common factor.

$16 x^{2} {(3 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.3.2.2

Rewrite the expression.

$16 x^{2} {(3 - x)}^{1} = 0^{2}$

Step 2.2.2.1.4

Simplify.

$16 x^{2} (3 - x) = 0^{2}$

Step 2.2.2.1.5

Simplify by multiplying through.

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Step 2.2.2.1.5.1

Apply the distributive property.

$16 x^{2} \cdot 3 + 16 x^{2} (- x) = 0^{2}$

Step 2.2.2.1.5.2

Simplify the expression.

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Step 2.2.2.1.5.2.1

Multiply

$3$ by

$16$ .

$48 x^{2} + 16 x^{2} (- x) = 0^{2}$

Step 2.2.2.1.5.2.2

Rewrite using the commutative property of multiplication.

$48 x^{2} + 16 \cdot - 1 x^{2} x = 0^{2}$

Step 2.2.2.1.6

Simplify each term.

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Step 2.2.2.1.6.1

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 2.2.2.1.6.1.1

Move

$x$ .

$48 x^{2} + 16 \cdot - 1 (x \cdot x^{2}) = 0^{2}$

Step 2.2.2.1.6.1.2

Multiply

$x$ by

$x^{2}$ .

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Step 2.2.2.1.6.1.2.1

Raise

$x$ to the power of

$1$ .

$48 x^{2} + 16 \cdot - 1 (x^{1} x^{2}) = 0^{2}$

Step 2.2.2.1.6.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$48 x^{2} + 16 \cdot - 1 x^{1 + 2} = 0^{2}$

Step 2.2.2.1.6.1.3

Add

$1$ and

$2$ .

$48 x^{2} + 16 \cdot - 1 x^{3} = 0^{2}$

Step 2.2.2.1.6.2

Multiply

$16$ by

$- 1$ .

$48 x^{2} - 16 x^{3} = 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$48 x^{2} - 16 x^{3} = 0$

Step 2.3

Solve for

$x$ .

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Step 2.3.1

Factor

$16 x^{2}$ out of

$48 x^{2} - 16 x^{3}$ .

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Step 2.3.1.1

Factor

$16 x^{2}$ out of

$48 x^{2}$ .

$16 x^{2} (3) - 16 x^{3} = 0$

Step 2.3.1.2

Factor

$16 x^{2}$ out of

$- 16 x^{3}$ .

$16 x^{2} (3) + 16 x^{2} (- x) = 0$

Step 2.3.1.3

Factor

$16 x^{2}$ out of

$16 x^{2} (3) + 16 x^{2} (- x)$ .

$16 x^{2} (3 - x) = 0$

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

$0$ .

$x^{2} = 0$

$3 - x = 0$

Step 2.3.3

Set

$x^{2}$ equal to

$0$ and solve for

$x$ .

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Step 2.3.3.1

Set

$x^{2}$ equal to

$0$ .

$x^{2} = 0$

Step 2.3.3.2

Solve

$x^{2} = 0$ for

$x$ .

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Step 2.3.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.3.3.2.2

Simplify

$\pm \sqrt{0}$ .

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Step 2.3.3.2.2.1

Rewrite

$0$ as

$0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.3.2.2.2

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

$x = \pm 0$

Step 2.3.3.2.2.3

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 2.3.4

Set

$3 - x$ equal to

$0$ and solve for

$x$ .

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Step 2.3.4.1

Set

$3 - x$ equal to

$0$ .

$3 - x = 0$

Step 2.3.4.2

Solve

$3 - x = 0$ for

$x$ .

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Step 2.3.4.2.1

Subtract

$3$ from both sides of the equation.

$- x = - 3$

Step 2.3.4.2.2

Divide each term in

$- x = - 3$ by

$- 1$ and simplify.

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Step 2.3.4.2.2.1

Divide each term in

$- x = - 3$ by

$- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

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.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 2.3.4.2.2.2.2

Divide

$x$ by

$1$ .

$x = \frac{- 3}{- 1}$

Step 2.3.4.2.2.3

Simplify the right side.

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Step 2.3.4.2.2.3.1

Divide

$- 3$ by

$- 1$ .

$x = 3$

Step 2.3.5

The final solution is all the values that make

$16 x^{2} (3 - x) = 0$ true.

$x = 0, 3$

Step 3