Algebra Examples

$\sqrt{3 x + 18} = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{3 x + 18} - x = 0$

Step 2

Factor $3$ out of $3 x + 18$ .

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Step 2.1

Factor $3$ out of $3 x$ .

$\sqrt{3 (x) + 18} - x = 0$

Step 2.2

Factor $3$ out of $18$ .

$\sqrt{3 x + 3 \cdot 6} - x = 0$

Step 2.3

Factor $3$ out of $3 x + 3 \cdot 6$ .

$\sqrt{3 (x + 6)} - x = 0$

Step 3

Add $x$ to both sides of the equation.

$\sqrt{3 (x + 6)} = x$

Step 4

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

${\sqrt{3 (x + 6)}}^{2} = x^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (x + 6)}$ as ${(3 (x + 6))}^{\frac{1}{2}}$ .

${({(3 (x + 6))}^{\frac{1}{2}})}^{2} = x^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${({(3 (x + 6))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Multiply the exponents in ${({(3 (x + 6))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1.1

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

${(3 (x + 6))}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

${(3 (x + 6))}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

${(3 (x + 6))}^{1} = x^{2}$

Step 5.2.1.2

Apply the distributive property.

${(3 x + 3 \cdot 6)}^{1} = x^{2}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $3$ by $6$ .

${(3 x + 18)}^{1} = x^{2}$

Step 5.2.1.3.2

Simplify.

$3 x + 18 = x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

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

$3 x + 18 - x^{2} = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Factor $- 1$ out of $3 x + 18 - x^{2}$ .

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Step 6.2.1.1

Reorder the expression.

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Step 6.2.1.1.1

Move $18$ .

$3 x - x^{2} + 18 = 0$

Step 6.2.1.1.2

Reorder $3 x$ and $- x^{2}$ .

$- x^{2} + 3 x + 18 = 0$

Step 6.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 3 x + 18 = 0$

Step 6.2.1.3

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) + 18 = 0$

Step 6.2.1.4

Rewrite $18$ as $- 1 (- 18)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot - 18 = 0$

Step 6.2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot - 18 = 0$

Step 6.2.1.6

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (- 18)$ .

$- (x^{2} - 3 x - 18) = 0$

Step 6.2.2

Factor.

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Step 6.2.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 6.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 6.2.2.1.2

Write the factored form using these integers.

$- ((x - 6) (x + 3)) = 0$

Step 6.2.2.2

Remove unnecessary parentheses.

$- (x - 6) (x + 3) = 0$

Step 6.3

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

$x - 6 = 0$

$x + 3 = 0$

Step 6.4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 6.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 6.5

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

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Step 6.5.1

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

$x + 3 = 0$

Step 6.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.6

The final solution is all the values that make $- (x - 6) (x + 3) = 0$ true.

$x = 6, - 3$

Step 7

Exclude the solutions that do not make $\sqrt{3 (x + 6)} - x = 0$ true.

$x = 6$