Algebra Examples

{(x - 0.1)}^{2} + {(2.2 - y)}^{2} = 25

${(x - 0.1)}^{2} + {(2.2 - y)}^{2} = 25$

Step 1

Simplify the left side

{(x - 0.1)}^{2} + {(2.2 - y)}^{2}

${(x - 0.1)}^{2} + {(2.2 - y)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x - 0.1)}^{2}

${(x - 0.1)}^{2}$ as

(x - 0.1) (x - 0.1)

$(x - 0.1) (x - 0.1)$ .

(x - 0.1) (x - 0.1) + {(2.2 - y)}^{2} = 25

$(x - 0.1) (x - 0.1) + {(2.2 - y)}^{2} = 25$

Step 1.1.2

Expand

(x - 0.1) (x - 0.1)

$(x - 0.1) (x - 0.1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

x (x - 0.1) - 0.1 (x - 0.1) + {(2.2 - y)}^{2} = 25

$x (x - 0.1) - 0.1 (x - 0.1) + {(2.2 - y)}^{2} = 25$

Step 1.1.2.2

Apply the distributive property.

x \cdot x + x \cdot - 0.1 - 0.1 (x - 0.1) + {(2.2 - y)}^{2} = 25

$x \cdot x + x \cdot - 0.1 - 0.1 (x - 0.1) + {(2.2 - y)}^{2} = 25$

Step 1.1.2.3

Apply the distributive property.

x \cdot x + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25

$x \cdot x + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25$

x \cdot x + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25

$x \cdot x + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25$

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

$x$ by

x

$x$ .

x^{2} + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25

$x^{2} + x \cdot - 0.1 - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25$

Step 1.1.3.1.2

Move

- 0.1

$- 0.1$ to the left of

x

$x$ .

x^{2} - 0.1 \cdot x - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25

$x^{2} - 0.1 \cdot x - 0.1 x - 0.1 \cdot - 0.1 + {(2.2 - y)}^{2} = 25$

Step 1.1.3.1.3

Multiply

- 0.1

$- 0.1$ by

- 0.1

$- 0.1$ .

x^{2} - 0.1 x - 0.1 x + 0.01 + {(2.2 - y)}^{2} = 25

$x^{2} - 0.1 x - 0.1 x + 0.01 + {(2.2 - y)}^{2} = 25$

x^{2} - 0.1 x - 0.1 x + 0.01 + {(2.2 - y)}^{2} = 25

$x^{2} - 0.1 x - 0.1 x + 0.01 + {(2.2 - y)}^{2} = 25$

Step 1.1.3.2

Subtract

0.1 x

$0.1 x$ from

- 0.1 x

$- 0.1 x$ .

x^{2} - 0.2 x + 0.01 + {(2.2 - y)}^{2} = 25

$x^{2} - 0.2 x + 0.01 + {(2.2 - y)}^{2} = 25$

x^{2} - 0.2 x + 0.01 + {(2.2 - y)}^{2} = 25

$x^{2} - 0.2 x + 0.01 + {(2.2 - y)}^{2} = 25$

Step 1.1.4

Rewrite

{(2.2 - y)}^{2}

${(2.2 - y)}^{2}$ as

(2.2 - y) (2.2 - y)

$(2.2 - y) (2.2 - y)$ .

x^{2} - 0.2 x + 0.01 + (2.2 - y) (2.2 - y) = 25

$x^{2} - 0.2 x + 0.01 + (2.2 - y) (2.2 - y) = 25$

Step 1.1.5

Expand

(2.2 - y) (2.2 - y)

$(2.2 - y) (2.2 - y)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

x^{2} - 0.2 x + 0.01 + 2.2 (2.2 - y) - y (2.2 - y) = 25

$x^{2} - 0.2 x + 0.01 + 2.2 (2.2 - y) - y (2.2 - y) = 25$

Step 1.1.5.2

Apply the distributive property.

x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y (2.2 - y) = 25

$x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y (2.2 - y) = 25$

Step 1.1.5.3

Apply the distributive property.

x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25

$x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25$

x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25

$x^{2} - 0.2 x + 0.01 + 2.2 \cdot 2.2 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25$

Step 1.1.6

Simplify and combine like terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

Multiply

2.2

$2.2$ by

2.2

$2.2$ .

x^{2} - 0.2 x + 0.01 + 4.84 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 + 2.2 (- y) - y \cdot 2.2 - y (- y) = 25$

Step 1.1.6.1.2

Multiply

- 1

$- 1$ by

2.2

$2.2$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - y \cdot 2.2 - y (- y) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - y \cdot 2.2 - y (- y) = 25$

Step 1.1.6.1.3

Multiply

2.2

$2.2$ by

- 1

$- 1$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - y (- y) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - y (- y) = 25$

Step 1.1.6.1.4

Rewrite using the commutative property of multiplication.

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y \cdot y) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y \cdot y) = 25$

Step 1.1.6.1.5

Multiply

y

$y$ by

y

$y$ by adding the exponents.

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Step 1.1.6.1.5.1

Move

y

$y$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 (y \cdot y)) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 (y \cdot y)) = 25$

Step 1.1.6.1.5.2

Multiply

y

$y$ by

y

$y$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y^{2}) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y^{2}) = 25$

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y^{2}) = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y - 1 \cdot (- 1 y^{2}) = 25$

Step 1.1.6.1.6

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + 1 y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + 1 y^{2} = 25$

Step 1.1.6.1.7

Multiply

y^{2}

$y^{2}$ by

1

$1$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + y^{2} = 25$

x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 2.2 y - 2.2 y + y^{2} = 25$

Step 1.1.6.2

Subtract

2.2 y

$2.2 y$ from

- 2.2 y

$- 2.2 y$ .

x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25$

x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25$

x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25

$x^{2} - 0.2 x + 0.01 + 4.84 - 4.4 y + y^{2} = 25$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add

0.01

$0.01$ and

4.84

$4.84$ .

x^{2} - 0.2 x + 4.85 - 4.4 y + y^{2} = 25

$x^{2} - 0.2 x + 4.85 - 4.4 y + y^{2} = 25$

Step 1.2.2

Move

4.85

$4.85$ .

x^{2} - 0.2 x - 4.4 y + y^{2} + 4.85 = 25

$x^{2} - 0.2 x - 4.4 y + y^{2} + 4.85 = 25$

Step 1.2.3

Move

- 4.4 y

$- 4.4 y$ .

x^{2} - 0.2 x + y^{2} - 4.4 y + 4.85 = 25

$x^{2} - 0.2 x + y^{2} - 4.4 y + 4.85 = 25$

Step 1.2.4

Move

- 0.2 x

$- 0.2 x$ .

x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25

$x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25$

x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25

$x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25$

x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25

$x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 = 25$

Step 2

Subtract

25

$25$ from both sides of the equation.

x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 - 25 = 0

$x^{2} + y^{2} - 0.2 x - 4.4 y + 4.85 - 25 = 0$

Step 3

Subtract

25

$25$ from

4.85

$4.85$ .

x^{2} + y^{2} - 0.2 x - 4.4 y - 20.15 = 0

$x^{2} + y^{2} - 0.2 x - 4.4 y - 20.15 = 0$

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