Algebra Examples

$\frac{1}{x} + 6 = \frac{5}{\sqrt{x}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{5}{\sqrt{x}} = \frac{1}{x} + 6$

Step 2

Multiply both sides by $\sqrt{x}$ .

$\frac{5}{\sqrt{x}} \sqrt{x} = (\frac{1}{x} + 6) \sqrt{x}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $\sqrt{x}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{5}{\sqrt{x}} \sqrt{x} = (\frac{1}{x} + 6) \sqrt{x}$

Step 3.1.1.2

Rewrite the expression.

$5 = (\frac{1}{x} + 6) \sqrt{x}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $(\frac{1}{x} + 6) \sqrt{x}$ .

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Step 3.2.1.1

Apply the distributive property.

$5 = \frac{1}{x} \sqrt{x} + 6 \sqrt{x}$

Step 3.2.1.2

Combine $\frac{1}{x}$ and $\sqrt{x}$ .

$5 = \frac{\sqrt{x}}{x} + 6 \sqrt{x}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{\sqrt{x}}{x} + 6 \sqrt{x} = 5$ .

$\frac{\sqrt{x}}{x} + 6 \sqrt{x} = 5$

Step 4.2

Solve for $\sqrt{x}$ .

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Step 4.2.1

Find the LCD of the terms in the equation.

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Step 4.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 4.2.1.2

The LCM of one and any expression is the expression.

$x$

Step 4.2.2

Multiply each term in $\frac{\sqrt{x}}{x} + 6 \sqrt{x} = 5$ by $x$ to eliminate the fractions.

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Step 4.2.2.1

Multiply each term in $\frac{\sqrt{x}}{x} + 6 \sqrt{x} = 5$ by $x$ .

$\frac{\sqrt{x}}{x} x + 6 \sqrt{x} x = 5 x$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $x$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{x}}{x} x + 6 \sqrt{x} x = 5 x$

Step 4.2.2.2.1.2

Rewrite the expression.

$\sqrt{x} + 6 \sqrt{x} x = 5 x$

Step 4.2.3

Solve the equation.

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Step 4.2.3.1

Factor $\sqrt{x}$ out of $\sqrt{x} + 6 \sqrt{x} x$ .

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Step 4.2.3.1.1

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

$\sqrt{x} + 6 \sqrt{x} x = 5 x$

Step 4.2.3.1.2

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

$\sqrt{x} \cdot 1 + 6 \sqrt{x} x = 5 x$

Step 4.2.3.1.3

Factor $\sqrt{x}$ out of $6 \sqrt{x} x$ .

$\sqrt{x} \cdot 1 + \sqrt{x} (6 x) = 5 x$

Step 4.2.3.1.4

Factor $\sqrt{x}$ out of $\sqrt{x} \cdot 1 + \sqrt{x} (6 x)$ .

$\sqrt{x} (1 + 6 x) = 5 x$

Step 4.2.3.2

Divide each term in $\sqrt{x} (1 + 6 x) = 5 x$ by $1 + 6 x$ and simplify.

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Step 4.2.3.2.1

Divide each term in $\sqrt{x} (1 + 6 x) = 5 x$ by $1 + 6 x$ .

$\frac{\sqrt{x} (1 + 6 x)}{1 + 6 x} = \frac{5 x}{1 + 6 x}$

Step 4.2.3.2.2

Simplify the left side.

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Step 4.2.3.2.2.1

Cancel the common factor of $1 + 6 x$ .

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Step 4.2.3.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{x} (1 + 6 x)}{1 + 6 x} = \frac{5 x}{1 + 6 x}$

Step 4.2.3.2.2.1.2

Divide $\sqrt{x}$ by $1$ .

$\sqrt{x} = \frac{5 x}{1 + 6 x}$

Step 4.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4

Simplify each side of the equation.

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Step 4.4.1

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

${(x^{\frac{1}{2}})}^{2} = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4.2.1.2

Simplify.

$x = {(\frac{5 x}{1 + 6 x})}^{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Simplify ${(\frac{5 x}{1 + 6 x})}^{2}$ .

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Step 4.4.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.4.3.1.1.1

Apply the product rule to $\frac{5 x}{1 + 6 x}$ .

$x = \frac{{(5 x)}^{2}}{{(1 + 6 x)}^{2}}$

Step 4.4.3.1.1.2

Apply the product rule to $5 x$ .

$x = \frac{5^{2} x^{2}}{{(1 + 6 x)}^{2}}$

Step 4.4.3.1.2

Raise $5$ to the power of $2$ .

$x = \frac{25 x^{2}}{{(1 + 6 x)}^{2}}$

Step 4.5

Solve for $x$ .

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Step 4.5.1

Find the LCD of the terms in the equation.

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Step 4.5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, {(1 + 6 x)}^{2}$

Step 4.5.1.2

The LCM of one and any expression is the expression.

${(1 + 6 x)}^{2}$

Step 4.5.2

Multiply each term in $x = \frac{25 x^{2}}{{(1 + 6 x)}^{2}}$ by ${(1 + 6 x)}^{2}$ to eliminate the fractions.

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Step 4.5.2.1

Multiply each term in $x = \frac{25 x^{2}}{{(1 + 6 x)}^{2}}$ by ${(1 + 6 x)}^{2}$ .

$x {(1 + 6 x)}^{2} = \frac{25 x^{2}}{{(1 + 6 x)}^{2}} {(1 + 6 x)}^{2}$

Step 4.5.2.2

Simplify the right side.

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Step 4.5.2.2.1

Cancel the common factor of ${(1 + 6 x)}^{2}$ .

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Step 4.5.2.2.1.1

Cancel the common factor.

$x {(1 + 6 x)}^{2} = \frac{25 x^{2}}{{(1 + 6 x)}^{2}} {(1 + 6 x)}^{2}$

Step 4.5.2.2.1.2

Rewrite the expression.

$x {(1 + 6 x)}^{2} = 25 x^{2}$

Step 4.5.3

Solve the equation.

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Step 4.5.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.5.3.1.1

Subtract $25 x^{2}$ from both sides of the equation.

$x {(1 + 6 x)}^{2} - 25 x^{2} = 0$

Step 4.5.3.1.2

Simplify each term.

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Step 4.5.3.1.2.1

Rewrite ${(1 + 6 x)}^{2}$ as $(1 + 6 x) (1 + 6 x)$ .

$x ((1 + 6 x) (1 + 6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.2

Expand $(1 + 6 x) (1 + 6 x)$ using the FOIL Method.

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Step 4.5.3.1.2.2.1

Apply the distributive property.

$x (1 (1 + 6 x) + 6 x (1 + 6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.2.2

Apply the distributive property.

$x (1 \cdot 1 + 1 (6 x) + 6 x (1 + 6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.2.3

Apply the distributive property.

$x (1 \cdot 1 + 1 (6 x) + 6 x \cdot 1 + 6 x (6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.3

Simplify and combine like terms.

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Step 4.5.3.1.2.3.1

Simplify each term.

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Step 4.5.3.1.2.3.1.1

Multiply $1$ by $1$ .

$x (1 + 1 (6 x) + 6 x \cdot 1 + 6 x (6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.2

Multiply $6 x$ by $1$ .

$x (1 + 6 x + 6 x \cdot 1 + 6 x (6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.3

Multiply $6$ by $1$ .

$x (1 + 6 x + 6 x + 6 x (6 x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.4

Rewrite using the commutative property of multiplication.

$x (1 + 6 x + 6 x + 6 \cdot 6 x \cdot x) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.5.3.1.2.3.1.5.1

Move $x$ .

$x (1 + 6 x + 6 x + 6 \cdot 6 (x \cdot x)) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.5.2

Multiply $x$ by $x$ .

$x (1 + 6 x + 6 x + 6 \cdot 6 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.1.6

Multiply $6$ by $6$ .

$x (1 + 6 x + 6 x + 36 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.3.2

Add $6 x$ and $6 x$ .

$x (1 + 12 x + 36 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.4

Apply the distributive property.

$x \cdot 1 + x (12 x) + x (36 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.5

Simplify.

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Step 4.5.3.1.2.5.1

Multiply $x$ by $1$ .

$x + x (12 x) + x (36 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.5.2

Rewrite using the commutative property of multiplication.

$x + 12 x \cdot x + x (36 x^{2}) - 25 x^{2} = 0$

Step 4.5.3.1.2.5.3

Rewrite using the commutative property of multiplication.

$x + 12 x \cdot x + 36 x \cdot x^{2} - 25 x^{2} = 0$

Step 4.5.3.1.2.6

Simplify each term.

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Step 4.5.3.1.2.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.5.3.1.2.6.1.1

Move $x$ .

$x + 12 (x \cdot x) + 36 x \cdot x^{2} - 25 x^{2} = 0$

Step 4.5.3.1.2.6.1.2

Multiply $x$ by $x$ .

$x + 12 x^{2} + 36 x \cdot x^{2} - 25 x^{2} = 0$

Step 4.5.3.1.2.6.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.5.3.1.2.6.2.1

Move $x^{2}$ .

$x + 12 x^{2} + 36 (x^{2} x) - 25 x^{2} = 0$

Step 4.5.3.1.2.6.2.2

Multiply $x^{2}$ by $x$ .

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Step 4.5.3.1.2.6.2.2.1

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

$x + 12 x^{2} + 36 (x^{2} x^{1}) - 25 x^{2} = 0$

Step 4.5.3.1.2.6.2.2.2

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

$x + 12 x^{2} + 36 x^{2 + 1} - 25 x^{2} = 0$

Step 4.5.3.1.2.6.2.3

Add $2$ and $1$ .

$x + 12 x^{2} + 36 x^{3} - 25 x^{2} = 0$

Step 4.5.3.1.3

Subtract $25 x^{2}$ from $12 x^{2}$ .

$x + 36 x^{3} - 13 x^{2} = 0$

Step 4.5.3.2

Factor the left side of the equation.

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Step 4.5.3.2.1

Factor $x$ out of $x + 36 x^{3} - 13 x^{2}$ .

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Step 4.5.3.2.1.1

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

$x + 36 x^{3} - 13 x^{2} = 0$

Step 4.5.3.2.1.2

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

$x \cdot 1 + 36 x^{3} - 13 x^{2} = 0$

Step 4.5.3.2.1.3

Factor $x$ out of $36 x^{3}$ .

$x \cdot 1 + x (36 x^{2}) - 13 x^{2} = 0$

Step 4.5.3.2.1.4

Factor $x$ out of $- 13 x^{2}$ .

$x \cdot 1 + x (36 x^{2}) + x (- 13 x) = 0$

Step 4.5.3.2.1.5

Factor $x$ out of $x \cdot 1 + x (36 x^{2})$ .

$x \cdot (1 + 36 x^{2}) + x (- 13 x) = 0$

Step 4.5.3.2.1.6

Factor $x$ out of $x \cdot (1 + 36 x^{2}) + x (- 13 x)$ .

$x (1 + 36 x^{2} - 13 x) = 0$

Step 4.5.3.2.2

Factor.

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Step 4.5.3.2.2.1

Factor by grouping.

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Step 4.5.3.2.2.1.1

Reorder terms.

$x (36 x^{2} - 13 x + 1) = 0$

Step 4.5.3.2.2.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 36 \cdot 1 = 36$ and whose sum is $b = - 13$ .

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Step 4.5.3.2.2.1.2.1

Factor $- 13$ out of $- 13 x$ .

$x (36 x^{2} - 13 x + 1) = 0$

Step 4.5.3.2.2.1.2.2

Rewrite $- 13$ as $- 4$ plus $- 9$

$x (36 x^{2} + (- 4 - 9) x + 1) = 0$

Step 4.5.3.2.2.1.2.3

Apply the distributive property.

$x (36 x^{2} - 4 x - 9 x + 1) = 0$

Step 4.5.3.2.2.1.3

Factor out the greatest common factor from each group.

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Step 4.5.3.2.2.1.3.1

Group the first two terms and the last two terms.

$x ((36 x^{2} - 4 x) - 9 x + 1) = 0$

Step 4.5.3.2.2.1.3.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x (9 x - 1) - (9 x - 1)) = 0$

Step 4.5.3.2.2.1.4

Factor the polynomial by factoring out the greatest common factor, $9 x - 1$ .

$x ((9 x - 1) (4 x - 1)) = 0$

Step 4.5.3.2.2.2

Remove unnecessary parentheses.

$x (9 x - 1) (4 x - 1) = 0$

Step 4.5.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$9 x - 1 = 0$

$4 x - 1 = 0$

Step 4.5.3.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5.3.5

Set $9 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.3.5.1

Set $9 x - 1$ equal to $0$ .

$9 x - 1 = 0$

Step 4.5.3.5.2

Solve $9 x - 1 = 0$ for $x$ .

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Step 4.5.3.5.2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 4.5.3.5.2.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 4.5.3.5.2.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.3.5.2.2.2

Simplify the left side.

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Step 4.5.3.5.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.5.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 4.5.3.6

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.3.6.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 4.5.3.6.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 4.5.3.6.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 4.5.3.6.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 4.5.3.6.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 4.5.3.6.2.2.2

Simplify the left side.

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Step 4.5.3.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.5.3.6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 4.5.3.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 4.5.3.7

The final solution is all the values that make $x (9 x - 1) (4 x - 1) = 0$ true.

$x = 0, \frac{1}{9}, \frac{1}{4}$

Step 5

Exclude the solutions that do not make $\frac{1}{x} + 6 = \frac{5}{\sqrt{x}}$ true.

$x = \frac{1}{9}, \frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{9}, \frac{1}{4}$

Decimal Form:

$x = 0.\overline{1}, 0.25$