Finite Math Examples

$2 x + \sqrt{7} = (3 - 2 x) \sqrt{7}$

Step 1

Simplify $(3 - 2 x) \sqrt{7}$ .

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Step 1.1

Rewrite.

$2 x + \sqrt{7} = 0 + 0 + (3 - 2 x) \sqrt{7}$

Step 1.2

Simplify by adding zeros.

$2 x + \sqrt{7} = (3 - 2 x) \sqrt{7}$

Step 1.3

Apply the distributive property.

$2 x + \sqrt{7} = 3 \sqrt{7} - 2 x \sqrt{7}$

Step 2

Add $2 x \sqrt{7}$ to both sides of the equation.

$2 x + \sqrt{7} + 2 x \sqrt{7} = 3 \sqrt{7}$

Step 3

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

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Step 3.1

Subtract $\sqrt{7}$ from both sides of the equation.

$2 x + 2 x \sqrt{7} = 3 \sqrt{7} - \sqrt{7}$

Step 3.2

Subtract $\sqrt{7}$ from $3 \sqrt{7}$ .

$2 x + 2 x \sqrt{7} = 2 \sqrt{7}$

Step 4

Factor $2 x$ out of $2 x + 2 x \sqrt{7}$ .

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Step 4.1

Factor $2 x$ out of $2 x$ .

$2 x (1) + 2 x \sqrt{7} = 2 \sqrt{7}$

Step 4.2

Factor $2 x$ out of $2 x \sqrt{7}$ .

$2 x (1) + 2 x (\sqrt{7}) = 2 \sqrt{7}$

Step 4.3

Factor $2 x$ out of $2 x (1) + 2 x (\sqrt{7})$ .

$2 x (1 + \sqrt{7}) = 2 \sqrt{7}$

Step 5

Divide each term in $2 x (1 + \sqrt{7}) = 2 \sqrt{7}$ by $2 + 2 \sqrt{7}$ and simplify.

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Step 5.1

Divide each term in $2 x (1 + \sqrt{7}) = 2 \sqrt{7}$ by $2 + 2 \sqrt{7}$ .

$\frac{2 x (1 + \sqrt{7})}{2 + 2 \sqrt{7}} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ and $2 + 2 \sqrt{7}$ .

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Step 5.2.1.1

Factor $2$ out of $2 x (1 + \sqrt{7})$ .

$\frac{2 (x (1 + \sqrt{7}))}{2 + 2 \sqrt{7}} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.1.2

Cancel the common factors.

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Step 5.2.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (x (1 + \sqrt{7}))}{2 \cdot 1 + 2 \sqrt{7}} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.1.2.2

Factor $2$ out of $2 \sqrt{7}$ .

$\frac{2 (x (1 + \sqrt{7}))}{2 \cdot 1 + 2 (\sqrt{7})} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.1.2.3

Factor $2$ out of $2 \cdot 1 + 2 (\sqrt{7})$ .

$\frac{2 (x (1 + \sqrt{7}))}{2 \cdot (1 + \sqrt{7})} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.1.2.4

Cancel the common factor.

$\frac{2 (x (1 + \sqrt{7}))}{2 \cdot (1 + \sqrt{7})} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.1.2.5

Rewrite the expression.

$\frac{x (1 + \sqrt{7})}{1 + \sqrt{7}} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.2

Cancel the common factor of $1 + \sqrt{7}$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{x (1 + \sqrt{7})}{1 + \sqrt{7}} = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.2.2.2

Divide $x$ by $1$ .

$x = \frac{2 \sqrt{7}}{2 + 2 \sqrt{7}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $2$ and $2 + 2 \sqrt{7}$ .

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Step 5.3.1.1

Factor $2$ out of $2 \sqrt{7}$ .

$x = \frac{2 (\sqrt{7})}{2 + 2 \sqrt{7}}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (\sqrt{7})}{2 \cdot 1 + 2 \sqrt{7}}$

Step 5.3.1.2.2

Factor $2$ out of $2 \sqrt{7}$ .

$x = \frac{2 (\sqrt{7})}{2 \cdot 1 + 2 (\sqrt{7})}$

Step 5.3.1.2.3

Factor $2$ out of $2 \cdot 1 + 2 (\sqrt{7})$ .

$x = \frac{2 (\sqrt{7})}{2 \cdot (1 + \sqrt{7})}$

Step 5.3.1.2.4

Cancel the common factor.

$x = \frac{2 \sqrt{7}}{2 \cdot (1 + \sqrt{7})}$

Step 5.3.1.2.5

Rewrite the expression.

$x = \frac{\sqrt{7}}{1 + \sqrt{7}}$

Step 5.3.2

Multiply $\frac{\sqrt{7}}{1 + \sqrt{7}}$ by $\frac{1 - \sqrt{7}}{1 - \sqrt{7}}$ .

$x = \frac{\sqrt{7}}{1 + \sqrt{7}} \cdot \frac{1 - \sqrt{7}}{1 - \sqrt{7}}$

Step 5.3.3

Multiply $\frac{\sqrt{7}}{1 + \sqrt{7}}$ by $\frac{1 - \sqrt{7}}{1 - \sqrt{7}}$ .

$x = \frac{\sqrt{7} (1 - \sqrt{7})}{(1 + \sqrt{7}) (1 - \sqrt{7})}$

Step 5.3.4

Expand the denominator using the FOIL method.

$x = \frac{\sqrt{7} (1 - \sqrt{7})}{1 - \sqrt{7} + \sqrt{7} - {\sqrt{7}}^{2}}$

Step 5.3.5

Simplify.

$x = \frac{\sqrt{7} (1 - \sqrt{7})}{- 6}$

Step 5.3.6

Apply the distributive property.

$x = \frac{\sqrt{7} \cdot 1 + \sqrt{7} (- \sqrt{7})}{- 6}$

Step 5.3.7

Multiply $\sqrt{7}$ by $1$ .

$x = \frac{\sqrt{7} + \sqrt{7} (- \sqrt{7})}{- 6}$

Step 5.3.8

Multiply $\sqrt{7} (- \sqrt{7})$ .

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Step 5.3.8.1

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{\sqrt{7} - ({\sqrt{7}}^{1} \sqrt{7})}{- 6}$

Step 5.3.8.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{\sqrt{7} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}{- 6}$

Step 5.3.8.3

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

$x = \frac{\sqrt{7} - {\sqrt{7}}^{1 + 1}}{- 6}$

Step 5.3.8.4

Add $1$ and $1$ .

$x = \frac{\sqrt{7} - {\sqrt{7}}^{2}}{- 6}$

Step 5.3.9

Simplify each term.

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Step 5.3.9.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.3.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \frac{\sqrt{7} - {(7^{\frac{1}{2}})}^{2}}{- 6}$

Step 5.3.9.1.2

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

$x = \frac{\sqrt{7} - 7^{\frac{1}{2} \cdot 2}}{- 6}$

Step 5.3.9.1.3

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

$x = \frac{\sqrt{7} - 7^{\frac{2}{2}}}{- 6}$

Step 5.3.9.1.4

Cancel the common factor of $2$ .

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Step 5.3.9.1.4.1

Cancel the common factor.

$x = \frac{\sqrt{7} - 7^{\frac{2}{2}}}{- 6}$

Step 5.3.9.1.4.2

Rewrite the expression.

$x = \frac{\sqrt{7} - 7^{1}}{- 6}$

Step 5.3.9.1.5

Evaluate the exponent.

$x = \frac{\sqrt{7} - 1 \cdot 7}{- 6}$

Step 5.3.9.2

Multiply $- 1$ by $7$ .

$x = \frac{\sqrt{7} - 7}{- 6}$

Step 5.3.10

Move the negative in front of the fraction.

$x = - \frac{\sqrt{7} - 7}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\sqrt{7} - 7}{6}$

Decimal Form:

$x = 0.72570811 \dots$