Algebra Examples

$x + x \sqrt{2} = 10$

Step 1

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

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Step 1.1

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

$x + x \sqrt{2} = 10$

Step 1.2

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

$x \cdot 1 + x \sqrt{2} = 10$

Step 1.3

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

$x \cdot 1 + x (\sqrt{2}) = 10$

Step 1.4

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

$x (1 + \sqrt{2}) = 10$

Step 2

Divide each term in $x (1 + \sqrt{2}) = 10$ by $1 + \sqrt{2}$ and simplify.

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Step 2.1

Divide each term in $x (1 + \sqrt{2}) = 10$ by $1 + \sqrt{2}$ .

$\frac{x (1 + \sqrt{2})}{1 + \sqrt{2}} = \frac{10}{1 + \sqrt{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$\frac{x (1 + \sqrt{2})}{1 + \sqrt{2}} = \frac{10}{1 + \sqrt{2}}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{1 + \sqrt{2}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $\frac{10}{1 + \sqrt{2}}$ by $\frac{1 - \sqrt{2}}{1 - \sqrt{2}}$ .

$x = \frac{10}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}}$

Step 2.3.2

Multiply $\frac{10}{1 + \sqrt{2}}$ by $\frac{1 - \sqrt{2}}{1 - \sqrt{2}}$ .

$x = \frac{10 (1 - \sqrt{2})}{(1 + \sqrt{2}) (1 - \sqrt{2})}$

Step 2.3.3

Expand the denominator using the FOIL method.

$x = \frac{10 (1 - \sqrt{2})}{1 - \sqrt{2} + \sqrt{2} - {\sqrt{2}}^{2}}$

Step 2.3.4

Simplify.

$x = \frac{10 (1 - \sqrt{2})}{- 1}$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Move the negative one from the denominator of $\frac{10 (1 - \sqrt{2})}{- 1}$ .

$x = - 1 \cdot (10 (1 - \sqrt{2}))$

Step 2.3.5.2

Rewrite $- 1 \cdot (10 (1 - \sqrt{2}))$ as $- (10 (1 - \sqrt{2}))$ .

$x = - (10 (1 - \sqrt{2}))$

Step 2.3.6

Apply the distributive property.

$x = - (10 \cdot 1 + 10 (- \sqrt{2}))$

Step 2.3.7

Multiply.

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Step 2.3.7.1

Multiply $10$ by $1$ .

$x = - (10 + 10 (- \sqrt{2}))$

Step 2.3.7.2

Multiply $- 1$ by $10$ .

$x = - (10 - 10 \sqrt{2})$

Step 2.3.8

Apply the distributive property.

$x = - 1 \cdot 10 - (- 10 \sqrt{2})$

Step 2.3.9

Multiply.

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Step 2.3.9.1

Multiply $- 1$ by $10$ .

$x = - 10 - (- 10 \sqrt{2})$

Step 2.3.9.2

Multiply $- 10$ by $- 1$ .

$x = - 10 + 10 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 10 + 10 \sqrt{2}$

Decimal Form:

$x = 4.14213562 \dots$