Precalculus Examples

${(x - \frac{15}{2})}^{2} + {(y - 7)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - \frac{15}{2})}^{2}$ as $(x - \frac{15}{2}) (x - \frac{15}{2})$ .

$(x - \frac{15}{2}) (x - \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.2

Expand $(x - \frac{15}{2}) (x - \frac{15}{2})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x (x - \frac{15}{2}) - \frac{15}{2} (x - \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.2.2

Apply the distributive property.

$x \cdot x + x (- \frac{15}{2}) - \frac{15}{2} (x - \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.2.3

Apply the distributive property.

$x \cdot x + x (- \frac{15}{2}) - \frac{15}{2} x - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x (- \frac{15}{2}) - \frac{15}{2} x - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3.1.2

Combine $x$ and $\frac{15}{2}$ .

$x^{2} - \frac{x \cdot 15}{2} - \frac{15}{2} x - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3.1.3

Move $15$ to the left of $x$ .

$x^{2} - \frac{15 \cdot x}{2} - \frac{15}{2} x - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3.1.4

Combine $x$ and $\frac{15}{2}$ .

$x^{2} - \frac{15 x}{2} - \frac{x \cdot 15}{2} - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3.1.5

Move $15$ to the left of $x$ .

$x^{2} - \frac{15 x}{2} - \frac{15 \cdot x}{2} - \frac{15}{2} (- \frac{15}{2}) + {(y - 7)}^{2}$

Step 1.3.1.6

Multiply $- \frac{15}{2} (- \frac{15}{2})$ .

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{15 x}{2} - \frac{15 x}{2} + 1 (\frac{15}{2}) \frac{15}{2} + {(y - 7)}^{2}$

Step 1.3.1.6.2

Multiply $\frac{15}{2}$ by $1$ .

$x^{2} - \frac{15 x}{2} - \frac{15 x}{2} + \frac{15}{2} \cdot \frac{15}{2} + {(y - 7)}^{2}$

Step 1.3.1.6.3

Multiply $\frac{15}{2}$ by $\frac{15}{2}$ .

$x^{2} - \frac{15 x}{2} - \frac{15 x}{2} + \frac{15 \cdot 15}{2 \cdot 2} + {(y - 7)}^{2}$

Step 1.3.1.6.4

Multiply $15$ by $15$ .

$x^{2} - \frac{15 x}{2} - \frac{15 x}{2} + \frac{225}{2 \cdot 2} + {(y - 7)}^{2}$

Step 1.3.1.6.5

Multiply $2$ by $2$ .

$x^{2} - \frac{15 x}{2} - \frac{15 x}{2} + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.3.2

Subtract $\frac{15 x}{2}$ from $- \frac{15 x}{2}$ .

$x^{2} - 2 \frac{15 x}{2} + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.4

Simplify each term.

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Step 1.4.1

Cancel the common factor of $2$ .

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Step 1.4.1.1

Factor $2$ out of $- 2$ .

$x^{2} + 2 (- 1) \frac{15 x}{2} + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.4.1.2

Cancel the common factor.

$x^{2} + 2 \cdot - 1 \frac{15 x}{2} + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.4.1.3

Rewrite the expression.

$x^{2} - 1 (15 x) + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.4.2

Multiply $15$ by $- 1$ .

$x^{2} - 15 x + \frac{225}{4} + {(y - 7)}^{2}$

Step 1.5

Rewrite ${(y - 7)}^{2}$ as $(y - 7) (y - 7)$ .

$x^{2} - 15 x + \frac{225}{4} + (y - 7) (y - 7)$

Step 1.6

Expand $(y - 7) (y - 7)$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

$x^{2} - 15 x + \frac{225}{4} + y (y - 7) - 7 (y - 7)$

Step 1.6.2

Apply the distributive property.

$x^{2} - 15 x + \frac{225}{4} + y \cdot y + y \cdot - 7 - 7 (y - 7)$

Step 1.6.3

Apply the distributive property.

$x^{2} - 15 x + \frac{225}{4} + y \cdot y + y \cdot - 7 - 7 y - 7 \cdot - 7$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $y$ by $y$ .

$x^{2} - 15 x + \frac{225}{4} + y^{2} + y \cdot - 7 - 7 y - 7 \cdot - 7$

Step 1.7.1.2

Move $- 7$ to the left of $y$ .

$x^{2} - 15 x + \frac{225}{4} + y^{2} - 7 \cdot y - 7 y - 7 \cdot - 7$

Step 1.7.1.3

Multiply $- 7$ by $- 7$ .

$x^{2} - 15 x + \frac{225}{4} + y^{2} - 7 y - 7 y + 49$

Step 1.7.2

Subtract $7 y$ from $- 7 y$ .

$x^{2} - 15 x + \frac{225}{4} + y^{2} - 14 y + 49$

Step 2

To write $49$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 15 x + y^{2} - 14 y + \frac{225}{4} + 49 \cdot \frac{4}{4}$

Step 3

Combine $49$ and $\frac{4}{4}$ .

$x^{2} - 15 x + y^{2} - 14 y + \frac{225}{4} + \frac{49 \cdot 4}{4}$

Step 4

Combine the numerators over the common denominator.

$x^{2} - 15 x + y^{2} - 14 y + \frac{225 + 49 \cdot 4}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $49$ by $4$ .

$x^{2} - 15 x + y^{2} - 14 y + \frac{225 + 196}{4}$

Step 5.2

Add $225$ and $196$ .

$x^{2} - 15 x + y^{2} - 14 y + \frac{421}{4}$

Step 6

Move $- 15 x$ .

$x^{2} + y^{2} - 15 x - 14 y + \frac{421}{4}$