Algebra Examples

$3 \sqrt{3 x} + 3 x = 0$

Step 1

Subtract $3 x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $3 x$ .

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

Step 3.2.1.2

Multiply $3$ by $3^{\frac{1}{2}}$ by adding the exponents.

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Step 3.2.1.2.1

Move $3^{\frac{1}{2}}$ .

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

Step 3.2.1.2.2

Multiply $3^{\frac{1}{2}}$ by $3$ .

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Step 3.2.1.2.2.1

Raise $3$ to the power of $1$ .

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

Step 3.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.1.2.3

Write $1$ as a fraction with a common denominator.

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

Step 3.2.1.2.4

Combine the numerators over the common denominator.

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

Step 3.2.1.2.5

Add $1$ and $2$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

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

Step 3.2.1.5

Raise $3$ to the power of $3$ .

$27 {(x^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Step 3.2.1.6

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

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Step 3.2.1.6.1

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

$27 x^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

Step 3.2.1.6.2

Cancel the common factor of $2$ .

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Step 3.2.1.6.2.1

Cancel the common factor.

$27 x^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

Step 3.2.1.6.2.2

Rewrite the expression.

$27 x^{1} = {(- 3 x)}^{2}$

Step 3.2.1.7

Simplify.

$27 x = {(- 3 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 3 x)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $- 3 x$ .

$27 x = {(- 3)}^{2} x^{2}$

Step 3.3.1.2

Raise $- 3$ to the power of $2$ .

$27 x = 9 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $9 x^{2}$ from both sides of the equation.

$27 x - 9 x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

$27 u - 9 u^{2} = 0$

Step 4.2.2

Factor $9 u$ out of $27 u - 9 u^{2}$ .

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Step 4.2.2.1

Factor $9 u$ out of $27 u$ .

$9 u (3) - 9 u^{2} = 0$

Step 4.2.2.2

Factor $9 u$ out of $- 9 u^{2}$ .

$9 u (3) + 9 u (- u) = 0$

Step 4.2.2.3

Factor $9 u$ out of $9 u (3) + 9 u (- u)$ .

$9 u (3 - u) = 0$

Step 4.2.3

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

$9 x (3 - x) = 0$

Step 4.3

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

$x = 0$

$3 - x = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $3 - x$ equal to $0$ and solve for $x$ .

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Step 4.5.1

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

$3 - x = 0$

Step 4.5.2

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

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Step 4.5.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 4.5.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 4.6

The final solution is all the values that make $9 x (3 - x) = 0$ true.

$x = 0, 3$

Step 5

Exclude the solutions that do not make $3 \sqrt{3 x} + 3 x = 0$ true.

$x = 0$