Algebra Examples

$5 x^{2} + x - \frac{1}{5} = 0$

Step 1

Add $\frac{1}{5}$ to both sides of the equation.

$5 x^{2} + x = \frac{1}{5}$

Step 2

Divide each term in $5 x^{2} + x = \frac{1}{5}$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 x^{2} + x = \frac{1}{5}$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{x}{5} = \frac{\frac{1}{5}}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{x}{5} = \frac{\frac{1}{5}}{5}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{x}{5} = \frac{\frac{1}{5}}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x^{2} + \frac{x}{5} = \frac{1}{5} \cdot \frac{1}{5}$

Step 2.3.2

Multiply $\frac{1}{5} \cdot \frac{1}{5}$ .

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Step 2.3.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$x^{2} + \frac{x}{5} = \frac{1}{5 \cdot 5}$

Step 2.3.2.2

Multiply $5$ by $5$ .

$x^{2} + \frac{x}{5} = \frac{1}{25}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{10})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{x}{5} + {(\frac{1}{10})}^{2} = \frac{1}{25} + {(\frac{1}{10})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{1}{10}$ .

$x^{2} + \frac{x}{5} + \frac{1^{2}}{10^{2}} = \frac{1}{25} + {(\frac{1}{10})}^{2}$

Step 5.1.1.2

One to any power is one.

$x^{2} + \frac{x}{5} + \frac{1}{10^{2}} = \frac{1}{25} + {(\frac{1}{10})}^{2}$

Step 5.1.1.3

Raise $10$ to the power of $2$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{25} + {(\frac{1}{10})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{1}{25} + {(\frac{1}{10})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{1}{10}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{25} + \frac{1^{2}}{10^{2}}$

Step 5.2.1.1.2

One to any power is one.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{25} + \frac{1}{10^{2}}$

Step 5.2.1.1.3

Raise $10$ to the power of $2$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{25} + \frac{1}{100}$

Step 5.2.1.2

To write $\frac{1}{25}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{25} \cdot \frac{4}{4} + \frac{1}{100}$

Step 5.2.1.3

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{1}{25}$ by $\frac{4}{4}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{4}{25 \cdot 4} + \frac{1}{100}$

Step 5.2.1.3.2

Multiply $25$ by $4$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{4}{100} + \frac{1}{100}$

Step 5.2.1.4

Simplify the expression.

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Step 5.2.1.4.1

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{4 + 1}{100}$

Step 5.2.1.4.2

Add $4$ and $1$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{5}{100}$

Step 5.2.1.5

Cancel the common factor of $5$ and $100$ .

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Step 5.2.1.5.1

Factor $5$ out of $5$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{5 (1)}{100}$

Step 5.2.1.5.2

Cancel the common factors.

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Step 5.2.1.5.2.1

Factor $5$ out of $100$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{5 \cdot 1}{5 \cdot 20}$

Step 5.2.1.5.2.2

Cancel the common factor.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{5 \cdot 1}{5 \cdot 20}$

Step 5.2.1.5.2.3

Rewrite the expression.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{20}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{1}{10})}^{2}$ .

${(x + \frac{1}{10})}^{2} = \frac{1}{20}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{1}{10} = \pm \sqrt{\frac{1}{20}}$

Step 7.2

Simplify $\pm \sqrt{\frac{1}{20}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{1}{20}}$ as $\frac{\sqrt{1}}{\sqrt{20}}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{1}}{\sqrt{20}}$

Step 7.2.2

Any root of $1$ is $1$ .

$x + \frac{1}{10} = \pm \frac{1}{\sqrt{20}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.2.3.1.1

Factor $4$ out of $20$ .

$x + \frac{1}{10} = \pm \frac{1}{\sqrt{4 (5)}}$

Step 7.2.3.1.2

Rewrite $4$ as $2^{2}$ .

$x + \frac{1}{10} = \pm \frac{1}{\sqrt{2^{2} \cdot 5}}$

Step 7.2.3.2

Pull terms out from under the radical.

$x + \frac{1}{10} = \pm \frac{1}{2 \sqrt{5}}$

Step 7.2.4

Multiply $\frac{1}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x + \frac{1}{10} = \pm \frac{1}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 7.2.5

Combine and simplify the denominator.

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Step 7.2.5.1

Multiply $\frac{1}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 7.2.5.2

Move $\sqrt{5}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 7.2.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 7.2.5.4

Raise $\sqrt{5}$ to the power of $1$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 7.2.5.5

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

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 7.2.5.6

Add $1$ and $1$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 7.2.5.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.2.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 7.2.5.7.2

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

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 7.2.5.7.3

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

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 7.2.5.7.4

Cancel the common factor of $2$ .

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Step 7.2.5.7.4.1

Cancel the common factor.

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 7.2.5.7.4.2

Rewrite the expression.

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \cdot 5^{1}}$

Step 7.2.5.7.5

Evaluate the exponent.

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{2 \cdot 5}$

Step 7.2.6

Multiply $2$ by $5$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{5}}{10}$

Step 7.3

Subtract $\frac{1}{10}$ from both sides of the equation.

$x = \pm \frac{\sqrt{5}}{10} - \frac{1}{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{5}}{10} - \frac{1}{10}$

Decimal Form:

$x = 0.12360679 \dots, - 0.32360679 \dots$