Algebra Examples

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

Step 1

Simplify the left side ${(x - \frac{1}{5})}^{2} + y^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - \frac{1}{5})}^{2}$ as $(x - \frac{1}{5}) (x - \frac{1}{5})$ .

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

Step 1.1.2

Expand $(x - \frac{1}{5}) (x - \frac{1}{5})$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$x (x - \frac{1}{5}) - \frac{1}{5} \cdot (x - \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x (- \frac{1}{5}) - \frac{1}{5} \cdot (x - \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x (- \frac{1}{5}) - \frac{1}{5} x - \frac{1}{5} \cdot (- \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x (- \frac{1}{5}) - \frac{1}{5} x - \frac{1}{5} \cdot (- \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.3.1.2

Combine $x$ and $\frac{1}{5}$ .

$x^{2} - \frac{x}{5} - \frac{1}{5} x - \frac{1}{5} \cdot (- \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.3.1.3

Combine $x$ and $\frac{1}{5}$ .

$x^{2} - \frac{x}{5} - \frac{x}{5} - \frac{1}{5} \cdot (- \frac{1}{5}) + y^{2} = \frac{1}{25}$

Step 1.1.3.1.4

Multiply $- \frac{1}{5} (- \frac{1}{5})$ .

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Step 1.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{x}{5} - \frac{x}{5} + 1 (\frac{1}{5}) \cdot \frac{1}{5} + y^{2} = \frac{1}{25}$

Step 1.1.3.1.4.2

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

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

Step 1.1.3.1.4.3

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

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

Step 1.1.3.1.4.4

Multiply $5$ by $5$ .

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

Step 1.1.3.2

Subtract $\frac{x}{5}$ from $- \frac{x}{5}$ .

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

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Combine $- 2$ and $\frac{x}{5}$ .

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

Step 1.1.4.2

Move the negative in front of the fraction.

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

Step 1.2

Reorder.

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Step 1.2.1

Move $\frac{1}{25}$ .

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

Step 1.2.2

Move $- \frac{2 x}{5}$ .

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

Step 2

Subtract $\frac{1}{25}$ from both sides of the equation.

$x^{2} + y^{2} - \frac{2 x}{5} + \frac{1}{25} - \frac{1}{25} = 0$

Step 3

Combine the opposite terms in $x^{2} + y^{2} - \frac{2 x}{5} + \frac{1}{25} - \frac{1}{25}$ .

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Step 3.1

Combine the numerators over the common denominator.

$x^{2} + y^{2} - \frac{2 x}{5} + \frac{1 - 1}{25} = 0$

Step 3.2

Subtract $1$ from $1$ .

$x^{2} + y^{2} - \frac{2 x}{5} + \frac{0}{25} = 0$

Step 3.3

Divide $0$ by $25$ .

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

Step 3.4

Add $x^{2} + y^{2} - \frac{2 x}{5}$ and $0$ .

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