Algebra Examples

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

Step 1

Set $4 x \sqrt{3 - x}$ equal to $0$ .

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

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify each side of the equation.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - x}$ as ${(3 - x)}^{\frac{1}{2}}$ .

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

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

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

Tap for more steps...

Step 2.2.2.1.1

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

Tap for more steps...

Step 2.2.2.1.1.1

Apply the product rule to $4 x {(3 - x)}^{\frac{1}{2}}$ .

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

Step 2.2.2.1.1.2

Apply the product rule to $4 x$ .

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

Step 2.2.2.1.2

Raise $4$ to the power of $2$ .

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

Step 2.2.2.1.3

Multiply the exponents in ${({(3 - x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 2.2.2.1.6.1

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

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 $x^{2}$ .

Tap for more steps...

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.

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 2.3.1

Factor $16 x^{2}$ out of $48 x^{2} - 16 x^{3}$ .

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 2.3.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.3.3.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

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 $\pm \sqrt{0}$ .

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 2.3.4.1

Set $3 - x$ equal to $0$ .

$3 - x = 0$

Step 2.3.4.2

Solve $3 - x = 0$ for $x$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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