Algebra Examples

$x \sqrt{2} - \sqrt{18} = x$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite $18$ as $3^{2} \cdot 2$ .

Tap for more steps...

Step 1.1.1

Factor $9$ out of $18$ .

$x \sqrt{2} - \sqrt{9 (2)} = x$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$x \sqrt{2} - \sqrt{3^{2} \cdot 2} = x$

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $3$ by $- 1$ .

$x \sqrt{2} - 3 \sqrt{2} = x$

Step 2

Subtract $x$ from both sides of the equation.

$x \sqrt{2} - 3 \sqrt{2} - x = 0$

Step 3

Add $3 \sqrt{2}$ to both sides of the equation.

$x \sqrt{2} - x = 3 \sqrt{2}$

Step 4

Factor $x$ out of $x \sqrt{2} - x$ .

Tap for more steps...

Step 4.1

Factor $x$ out of $x \sqrt{2}$ .

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

Step 4.2

Factor $x$ out of $- x$ .

$x (\sqrt{2}) + x \cdot - 1 = 3 \sqrt{2}$

Step 4.3

Factor $x$ out of $x (\sqrt{2}) + x \cdot - 1$ .

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

Step 5

Divide each term in $x (\sqrt{2} - 1) = 3 \sqrt{2}$ by $\sqrt{2} - 1$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $x (\sqrt{2} - 1) = 3 \sqrt{2}$ by $\sqrt{2} - 1$ .

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

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $\sqrt{2} - 1$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \sqrt{2}}{\sqrt{2} - 1}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{2} - 1}$ by $\frac{\sqrt{2} + 1}{\sqrt{2} + 1}$ .

$x = \frac{3 \sqrt{2}}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1}$

Step 5.3.2

Multiply $\frac{3 \sqrt{2}}{\sqrt{2} - 1}$ by $\frac{\sqrt{2} + 1}{\sqrt{2} + 1}$ .

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

Step 5.3.3

Expand the denominator using the FOIL method.

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

Step 5.3.4

Simplify.

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

Step 5.3.5

Divide $3 \sqrt{2} (\sqrt{2} + 1)$ by $1$ .

$x = 3 \sqrt{2} (\sqrt{2} + 1)$

Step 5.3.6

Apply the distributive property.

$x = 3 \sqrt{2} \sqrt{2} + 3 \sqrt{2} \cdot 1$

Step 5.3.7

Multiply $3 \sqrt{2} \sqrt{2}$ .

Tap for more steps...

Step 5.3.7.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.7.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.7.3

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

$x = 3 {\sqrt{2}}^{1 + 1} + 3 \sqrt{2} \cdot 1$

Step 5.3.7.4

Add $1$ and $1$ .

$x = 3 {\sqrt{2}}^{2} + 3 \sqrt{2} \cdot 1$

Step 5.3.8

Multiply $3$ by $1$ .

$x = 3 {\sqrt{2}}^{2} + 3 \sqrt{2}$

Step 5.3.9

Simplify each term.

Tap for more steps...

Step 5.3.9.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 5.3.9.1.1

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

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

Step 5.3.9.1.2

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

$x = 3 \cdot 2^{\frac{1}{2} \cdot 2} + 3 \sqrt{2}$

Step 5.3.9.1.3

Combine $\frac{1}{2}$ and $2$ .

$x = 3 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2}$

Step 5.3.9.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.9.1.4.1

Cancel the common factor.

$x = 3 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2}$

Step 5.3.9.1.4.2

Rewrite the expression.

$x = 3 \cdot 2^{1} + 3 \sqrt{2}$

Step 5.3.9.1.5

Evaluate the exponent.

$x = 3 \cdot 2 + 3 \sqrt{2}$

Step 5.3.9.2

Multiply $3$ by $2$ .

$x = 6 + 3 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 6 + 3 \sqrt{2}$

Decimal Form:

$x = 10.24264068 \dots$