Basic Math Examples

$\sqrt{2} z - 3 + z = 3$

Step 1

Move all terms not containing $z$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$\sqrt{2} z + z = 3 + 3$

Step 1.2

Add $3$ and $3$ .

$\sqrt{2} z + z = 6$

Step 2

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

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Step 2.1

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

$z \sqrt{2} + z = 6$

Step 2.2

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

$z \sqrt{2} + z = 6$

Step 2.3

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

$z \sqrt{2} + z \cdot 1 = 6$

Step 2.4

Factor $z$ out of $z \sqrt{2} + z \cdot 1$ .

$z (\sqrt{2} + 1) = 6$

Step 3

Divide each term in $z (\sqrt{2} + 1) = 6$ by $\sqrt{2} + 1$ and simplify.

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Step 3.1

Divide each term in $z (\sqrt{2} + 1) = 6$ by $\sqrt{2} + 1$ .

$\frac{z (\sqrt{2} + 1)}{\sqrt{2} + 1} = \frac{6}{\sqrt{2} + 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{z (\sqrt{2} + 1)}{\sqrt{2} + 1} = \frac{6}{\sqrt{2} + 1}$

Step 3.2.1.2

Divide $z$ by $1$ .

$z = \frac{6}{\sqrt{2} + 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $\frac{6}{\sqrt{2} + 1}$ by $\frac{\sqrt{2} - 1}{\sqrt{2} - 1}$ .

$z = \frac{6}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1}$

Step 3.3.2

Multiply $\frac{6}{\sqrt{2} + 1}$ by $\frac{\sqrt{2} - 1}{\sqrt{2} - 1}$ .

$z = \frac{6 (\sqrt{2} - 1)}{(\sqrt{2} + 1) (\sqrt{2} - 1)}$

Step 3.3.3

Expand the denominator using the FOIL method.

$z = \frac{6 (\sqrt{2} - 1)}{{\sqrt{2}}^{2} + \sqrt{2} \cdot - 1 + \sqrt{2} - 1}$

Step 3.3.4

Simplify.

$z = \frac{6 (\sqrt{2} - 1)}{1}$

Step 3.3.5

Divide $6 (\sqrt{2} - 1)$ by $1$ .

$z = 6 (\sqrt{2} - 1)$

Step 3.3.6

Apply the distributive property.

$z = 6 \sqrt{2} + 6 \cdot - 1$

Step 3.3.7

Multiply $6$ by $- 1$ .

$z = 6 \sqrt{2} - 6$

Step 4

The result can be shown in multiple forms.

Exact Form:

$z = 6 \sqrt{2} - 6$

Decimal Form:

$z = 2.48528137 \dots$