Trigonometry Examples

$9 {(x - 4)}^{2} + 16 {(y - 3)}^{2} = 144$

Step 1

Subtract $16 {(y - 3)}^{2}$ from both sides of the equation.

$9 {(x - 4)}^{2} = 144 - 16 {(y - 3)}^{2}$

Step 2

Simplify $144 - 16 {(y - 3)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$9 {(x - 4)}^{2} = 144 - 16 ((y - 3) (y - 3))$

Step 2.1.2

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$9 {(x - 4)}^{2} = 144 - 16 (y (y - 3) - 3 (y - 3))$

Step 2.1.2.2

Apply the distributive property.

$9 {(x - 4)}^{2} = 144 - 16 (y \cdot y + y \cdot - 3 - 3 (y - 3))$

Step 2.1.2.3

Apply the distributive property.

$9 {(x - 4)}^{2} = 144 - 16 (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3)$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $y$ by $y$ .

$9 {(x - 4)}^{2} = 144 - 16 (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3)$

Step 2.1.3.1.2

Move $- 3$ to the left of $y$ .

$9 {(x - 4)}^{2} = 144 - 16 (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3)$

Step 2.1.3.1.3

Multiply $- 3$ by $- 3$ .

$9 {(x - 4)}^{2} = 144 - 16 (y^{2} - 3 y - 3 y + 9)$

Step 2.1.3.2

Subtract $3 y$ from $- 3 y$ .

$9 {(x - 4)}^{2} = 144 - 16 (y^{2} - 6 y + 9)$

Step 2.1.4

Apply the distributive property.

$9 {(x - 4)}^{2} = 144 - 16 y^{2} - 16 (- 6 y) - 16 \cdot 9$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $- 6$ by $- 16$ .

$9 {(x - 4)}^{2} = 144 - 16 y^{2} + 96 y - 16 \cdot 9$

Step 2.1.5.2

Multiply $- 16$ by $9$ .

$9 {(x - 4)}^{2} = 144 - 16 y^{2} + 96 y - 144$

Step 2.2

Combine the opposite terms in $144 - 16 y^{2} + 96 y - 144$ .

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Step 2.2.1

Subtract $144$ from $144$ .

$9 {(x - 4)}^{2} = - 16 y^{2} + 96 y + 0$

Step 2.2.2

Add $- 16 y^{2} + 96 y$ and $0$ .

$9 {(x - 4)}^{2} = - 16 y^{2} + 96 y$

Step 3

Divide each term in $9 {(x - 4)}^{2} = - 16 y^{2} + 96 y$ by $9$ and simplify.

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Step 3.1

Divide each term in $9 {(x - 4)}^{2} = - 16 y^{2} + 96 y$ by $9$ .

$\frac{9 {(x - 4)}^{2}}{9} = \frac{- 16 y^{2}}{9} + \frac{96 y}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 {(x - 4)}^{2}}{9} = \frac{- 16 y^{2}}{9} + \frac{96 y}{9}$

Step 3.2.1.2

Divide ${(x - 4)}^{2}$ by $1$ .

${(x - 4)}^{2} = \frac{- 16 y^{2}}{9} + \frac{96 y}{9}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

${(x - 4)}^{2} = - \frac{16 y^{2}}{9} + \frac{96 y}{9}$

Step 3.3.1.2

Cancel the common factor of $96$ and $9$ .

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Step 3.3.1.2.1

Factor $3$ out of $96 y$ .

${(x - 4)}^{2} = - \frac{16 y^{2}}{9} + \frac{3 (32 y)}{9}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $3$ out of $9$ .

${(x - 4)}^{2} = - \frac{16 y^{2}}{9} + \frac{3 (32 y)}{3 (3)}$

Step 3.3.1.2.2.2

Cancel the common factor.

${(x - 4)}^{2} = - \frac{16 y^{2}}{9} + \frac{3 (32 y)}{3 \cdot 3}$

Step 3.3.1.2.2.3

Rewrite the expression.

${(x - 4)}^{2} = - \frac{16 y^{2}}{9} + \frac{32 y}{3}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 4 = \pm \sqrt{- \frac{16 y^{2}}{9} + \frac{32 y}{3}}$

Step 5

Simplify $\pm \sqrt{- \frac{16 y^{2}}{9} + \frac{32 y}{3}}$ .

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Step 5.1

To write $\frac{32 y}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x - 4 = \pm \sqrt{- \frac{16 y^{2}}{9} + \frac{32 y}{3} \cdot \frac{3}{3}}$

Step 5.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{32 y}{3}$ by $\frac{3}{3}$ .

$x - 4 = \pm \sqrt{- \frac{16 y^{2}}{9} + \frac{32 y \cdot 3}{3 \cdot 3}}$

Step 5.2.2

Multiply $3$ by $3$ .

$x - 4 = \pm \sqrt{- \frac{16 y^{2}}{9} + \frac{32 y \cdot 3}{9}}$

Step 5.3

Combine the numerators over the common denominator.

$x - 4 = \pm \sqrt{\frac{- 16 y^{2} + 32 y \cdot 3}{9}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Factor $16 y$ out of $- 16 y^{2} + 32 y \cdot 3$ .

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Step 5.4.1.1

Factor $16 y$ out of $- 16 y^{2}$ .

$x - 4 = \pm \sqrt{\frac{16 y (- y) + 32 y \cdot 3}{9}}$

Step 5.4.1.2

Factor $16 y$ out of $32 y \cdot 3$ .

$x - 4 = \pm \sqrt{\frac{16 y (- y) + 16 y (2 \cdot 3)}{9}}$

Step 5.4.1.3

Factor $16 y$ out of $16 y (- y) + 16 y (2 \cdot 3)$ .

$x - 4 = \pm \sqrt{\frac{16 y (- y + 2 \cdot 3)}{9}}$

Step 5.4.2

Multiply $2$ by $3$ .

$x - 4 = \pm \sqrt{\frac{16 y (- y + 6)}{9}}$

Step 5.5

Rewrite $\frac{16 y (- y + 6)}{9}$ as ${(\frac{2^{2}}{3})}^{2} (y (- y + 6))$ .

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Step 5.5.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 y (- y + 6)$ .

$x - 4 = \pm \sqrt{\frac{{(2^{2})}^{2} (y (- y + 6))}{9}}$

Step 5.5.2

Factor the perfect power $3^{2}$ out of $9$ .

$x - 4 = \pm \sqrt{\frac{{(2^{2})}^{2} (y (- y + 6))}{3^{2} \cdot 1}}$

Step 5.5.3

Rearrange the fraction $\frac{{(2^{2})}^{2} (y (- y + 6))}{3^{2} \cdot 1}$ .

$x - 4 = \pm \sqrt{{(\frac{2^{2}}{3})}^{2} (y (- y + 6))}$

Step 5.6

Pull terms out from under the radical.

$x - 4 = \pm \frac{2^{2}}{3} \sqrt{y (- y + 6)}$

Step 5.7

Raise $2$ to the power of $2$ .

$x - 4 = \pm \frac{4}{3} \sqrt{y (- y + 6)}$

Step 5.8

Combine $\frac{4}{3}$ and $\sqrt{y (- y + 6)}$ .

$x - 4 = \pm \frac{4 \sqrt{y (- y + 6)}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 4 = \frac{4 \sqrt{y (- y + 6)}}{3}$

Step 6.2

Add $4$ to both sides of the equation.

$x = \frac{4 \sqrt{y (- y + 6)}}{3} + 4$

Step 6.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 4 = - \frac{4 \sqrt{y (- y + 6)}}{3}$

Step 6.4

Add $4$ to both sides of the equation.

$x = - \frac{4 \sqrt{y (- y + 6)}}{3} + 4$

Step 6.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4 \sqrt{y (- y + 6)}}{3} + 4$

$x = - \frac{4 \sqrt{y (- y + 6)}}{3} + 4$

$x = \frac{4 \sqrt{y (- y + 6)}}{3} + 4$

$x = - \frac{4 \sqrt{y (- y + 6)}}{3} + 4$