Precalculus Examples

$\frac{{(x - 5)}^{2}}{16} + \frac{y^{2}}{1} = 1$

Step 1

Simplify the left side $\frac{{(x - 5)}^{2}}{16} + \frac{y^{2}}{1}$ .

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Step 1.1

Divide $y^{2}$ by $1$ .

$\frac{{(x - 5)}^{2}}{16} + y^{2} = 1$

Step 1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{{(x - 5)}^{2}}{16} + y^{2} \cdot \frac{16}{16} = 1$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $y^{2}$ and $\frac{16}{16}$ .

$\frac{{(x - 5)}^{2}}{16} + \frac{y^{2} \cdot 16}{16} = 1$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{{(x - 5)}^{2} + y^{2} \cdot 16}{16} = 1$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

$\frac{(x - 5) (x - 5) + y^{2} \cdot 16}{16} = 1$

Step 1.4.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.4.2.1

Apply the distributive property.

$\frac{x (x - 5) - 5 (x - 5) + y^{2} \cdot 16}{16} = 1$

Step 1.4.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 - 5 (x - 5) + y^{2} \cdot 16}{16} = 1$

Step 1.4.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + y^{2} \cdot 16}{16} = 1$

Step 1.4.3

Simplify and combine like terms.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + y^{2} \cdot 16}{16} = 1$

Step 1.4.3.1.2

Move $- 5$ to the left of $x$ .

$\frac{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + y^{2} \cdot 16}{16} = 1$

Step 1.4.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{x^{2} - 5 x - 5 x + 25 + y^{2} \cdot 16}{16} = 1$

Step 1.4.3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{x^{2} - 10 x + 25 + y^{2} \cdot 16}{16} = 1$

Step 1.4.4

Move $16$ to the left of $y^{2}$ .

$\frac{x^{2} - 10 x + 25 + 16 y^{2}}{16} = 1$

Step 2

Multiply both sides by $16$ .

$\frac{x^{2} - 10 x + 25 + 16 y^{2}}{16} \cdot 16 = 1 \cdot 16$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{x^{2} - 10 x + 25 + 16 y^{2}}{16} \cdot 16$ .

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Step 3.1.1.1

Cancel the common factor of $16$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{x^{2} - 10 x + 25 + 16 y^{2}}{16} \cdot 16 = 1 \cdot 16$

Step 3.1.1.1.2

Rewrite the expression.

$x^{2} - 10 x + 25 + 16 y^{2} = 1 \cdot 16$

Step 3.1.1.2

Reorder.

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Step 3.1.1.2.1

Move $25$ .

$x^{2} - 10 x + 16 y^{2} + 25 = 1 \cdot 16$

Step 3.1.1.2.2

Move $- 10 x$ .

$x^{2} + 16 y^{2} - 10 x + 25 = 1 \cdot 16$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $16$ by $1$ .

$x^{2} + 16 y^{2} - 10 x + 25 = 16$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $16$ from both sides of the equation.

$x^{2} + 16 y^{2} - 10 x + 25 - 16 = 0$

Step 4.2

Subtract $16$ from $25$ .

$x^{2} + 16 y^{2} - 10 x + 9 = 0$