Find the Expanded Form ((x-12)^2)/14+((y+15)^2)/4=36

$\frac{{(x - 12)}^{2}}{14} + \frac{{(y + 15)}^{2}}{4} = 36$

Step 1

Simplify the left side $\frac{{(x - 12)}^{2}}{14} + \frac{{(y + 15)}^{2}}{4}$ .

To write $\frac{{(x - 12)}^{2}}{14}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(x - 12)}^{2}}{14} \cdot \frac{2}{2} + \frac{{(y + 15)}^{2}}{4} = 36$

To write $\frac{{(y + 15)}^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{{(x - 12)}^{2}}{14} \cdot \frac{2}{2} + \frac{{(y + 15)}^{2}}{4} \cdot \frac{7}{7} = 36$

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{{(x - 12)}^{2}}{14}$ by $\frac{2}{2}$ .

$\frac{{(x - 12)}^{2} \cdot 2}{14 \cdot 2} + \frac{{(y + 15)}^{2}}{4} \cdot \frac{7}{7} = 36$

Multiply $14$ by $2$ .

$\frac{{(x - 12)}^{2} \cdot 2}{28} + \frac{{(y + 15)}^{2}}{4} \cdot \frac{7}{7} = 36$

Multiply $\frac{{(y + 15)}^{2}}{4}$ by $\frac{7}{7}$ .

$\frac{{(x - 12)}^{2} \cdot 2}{28} + \frac{{(y + 15)}^{2} \cdot 7}{4 \cdot 7} = 36$

Multiply $4$ by $7$ .

$\frac{{(x - 12)}^{2} \cdot 2}{28} + \frac{{(y + 15)}^{2} \cdot 7}{28} = 36$

Combine the numerators over the common denominator.

$\frac{{(x - 12)}^{2} \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Simplify the numerator.

Rewrite ${(x - 12)}^{2}$ as $(x - 12) (x - 12)$ .

$\frac{(x - 12) (x - 12) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Expand $(x - 12) (x - 12)$ using the FOIL Method.

Apply the distributive property.

$\frac{(x (x - 12) - 12 (x - 12)) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 12 - 12 (x - 12)) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 12 - 12 x - 12 \cdot - 12) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot - 12 - 12 x - 12 \cdot - 12) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Move $- 12$ to the left of $x$ .

$\frac{(x^{2} - 12 \cdot x - 12 x - 12 \cdot - 12) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Multiply $- 12$ by $- 12$ .

$\frac{(x^{2} - 12 x - 12 x + 144) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Subtract $12 x$ from $- 12 x$ .

$\frac{(x^{2} - 24 x + 144) \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{x^{2} \cdot 2 - 24 x \cdot 2 + 144 \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Simplify.

Move $2$ to the left of $x^{2}$ .

$\frac{2 \cdot x^{2} - 24 x \cdot 2 + 144 \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Multiply $2$ by $- 24$ .

$\frac{2 \cdot x^{2} - 48 x + 144 \cdot 2 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Multiply $144$ by $2$ .

$\frac{2 \cdot x^{2} - 48 x + 288 + {(y + 15)}^{2} \cdot 7}{28} = 36$

Rewrite ${(y + 15)}^{2}$ as $(y + 15) (y + 15)$ .

$\frac{2 x^{2} - 48 x + 288 + (y + 15) (y + 15) \cdot 7}{28} = 36$

Expand $(y + 15) (y + 15)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x^{2} - 48 x + 288 + (y (y + 15) + 15 (y + 15)) \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{2 x^{2} - 48 x + 288 + (y \cdot y + y \cdot 15 + 15 (y + 15)) \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{2 x^{2} - 48 x + 288 + (y \cdot y + y \cdot 15 + 15 y + 15 \cdot 15) \cdot 7}{28} = 36$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{2 x^{2} - 48 x + 288 + (y^{2} + y \cdot 15 + 15 y + 15 \cdot 15) \cdot 7}{28} = 36$

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

$\frac{2 x^{2} - 48 x + 288 + (y^{2} + 15 \cdot y + 15 y + 15 \cdot 15) \cdot 7}{28} = 36$

Multiply $15$ by $15$ .

$\frac{2 x^{2} - 48 x + 288 + (y^{2} + 15 y + 15 y + 225) \cdot 7}{28} = 36$

Add $15 y$ and $15 y$ .

$\frac{2 x^{2} - 48 x + 288 + (y^{2} + 30 y + 225) \cdot 7}{28} = 36$

Apply the distributive property.

$\frac{2 x^{2} - 48 x + 288 + y^{2} \cdot 7 + 30 y \cdot 7 + 225 \cdot 7}{28} = 36$

Simplify.

Move $7$ to the left of $y^{2}$ .

$\frac{2 x^{2} - 48 x + 288 + 7 \cdot y^{2} + 30 y \cdot 7 + 225 \cdot 7}{28} = 36$

Multiply $7$ by $30$ .

$\frac{2 x^{2} - 48 x + 288 + 7 \cdot y^{2} + 210 y + 225 \cdot 7}{28} = 36$

Multiply $225$ by $7$ .

$\frac{2 x^{2} - 48 x + 288 + 7 \cdot y^{2} + 210 y + 1575}{28} = 36$

Add $288$ and $1575$ .

$\frac{2 x^{2} - 48 x + 7 y^{2} + 210 y + 1863}{28} = 36$

Step 2

Multiply both sides by $28$ .

$\frac{2 x^{2} - 48 x + 7 y^{2} + 210 y + 1863}{28} \cdot 28 = 36 \cdot 28$

Step 3

Simplify.

Simplify the left side.

Simplify $\frac{2 x^{2} - 48 x + 7 y^{2} + 210 y + 1863}{28} \cdot 28$ .

Cancel the common factor of $28$ .

Cancel the common factor.

$\frac{2 x^{2} - 48 x + 7 y^{2} + 210 y + 1863}{28} \cdot 28 = 36 \cdot 28$

Rewrite the expression.

$2 x^{2} - 48 x + 7 y^{2} + 210 y + 1863 = 36 \cdot 28$

Move $- 48 x$ .

$2 x^{2} + 7 y^{2} - 48 x + 210 y + 1863 = 36 \cdot 28$

Simplify the right side.

Multiply $36$ by $28$ .

$2 x^{2} + 7 y^{2} - 48 x + 210 y + 1863 = 1008$

Step 4

Set the equation equal to zero.

Subtract $1008$ from both sides of the equation.

$2 x^{2} + 7 y^{2} - 48 x + 210 y + 1863 - 1008 = 0$

Subtract $1008$ from $1863$ .

$2 x^{2} + 7 y^{2} - 48 x + 210 y + 855 = 0$

Find the Expanded Form ((x-12)^2)/14+((y+15)^2)/4=36

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