Algebra Examples

${(y - 0)}^{2} = 9 - 9 \frac{{(x + 3)}^{2}}{16}$

Step 1

Multiply $- 1$ by $0$ .

${(y + 0)}^{2} = 9 - 9 \frac{{(x + 3)}^{2}}{16}$

Step 2

Add $y$ and $0$ .

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

Step 3

Combine $- 9$ and $\frac{{(x + 3)}^{2}}{16}$ .

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{9 + \frac{- 9 {(x + 3)}^{2}}{16}}$ .

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Step 5.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

Step 5.2

Combine $9$ and $\frac{16}{16}$ .

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Rewrite $\frac{9 \cdot 16 - 9 {(x + 3)}^{2}}{16}$ in a factored form.

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Step 5.4.1

Factor $9$ out of $9 \cdot 16 - 9 {(x + 3)}^{2}$ .

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Step 5.4.1.1

Factor $9$ out of $9 \cdot 16$ .

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

Step 5.4.1.2

Factor $9$ out of $- 9 {(x + 3)}^{2}$ .

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

Step 5.4.1.3

Factor $9$ out of $9 (16) + 9 (- {(x + 3)}^{2})$ .

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

Step 5.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 5.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x + 3$ .

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

Step 5.4.4

Simplify.

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Step 5.4.4.1

Add $4$ and $3$ .

$y = \pm \sqrt{\frac{9 ((x + 7) (4 - (x + 3)))}{16}}$

Step 5.4.4.2

Apply the distributive property.

$y = \pm \sqrt{\frac{9 ((x + 7) (4 - x - 1 \cdot 3))}{16}}$

Step 5.4.4.3

Multiply $- 1$ by $3$ .

$y = \pm \sqrt{\frac{9 ((x + 7) (4 - x - 3))}{16}}$

Step 5.4.4.4

Subtract $3$ from $4$ .

$y = \pm \sqrt{\frac{9 ((x + 7) (- x + 1))}{16}}$

$y = \pm \sqrt{\frac{9 (x + 7) (- x + 1)}{16}}$

Step 5.5

Rewrite $\frac{9 (x + 7) (- x + 1)}{16}$ as ${(\frac{3}{4})}^{2} ((x + 7) (- x + 1^{2}))$ .

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Step 5.5.1

Factor the perfect power $3^{2}$ out of $9 (x + 7) (- x + 1)$ .

$y = \pm \sqrt{\frac{3^{2} ((x + 7) (- x + 1^{2}))}{16}}$

Step 5.5.2

Factor the perfect power $4^{2}$ out of $16$ .

$y = \pm \sqrt{\frac{3^{2} ((x + 7) (- x + 1^{2}))}{4^{2} \cdot 1}}$

Step 5.5.3

Rearrange the fraction $\frac{3^{2} ((x + 7) (- x + 1^{2}))}{4^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{3}{4})}^{2} ((x + 7) (- x + 1^{2}))}$

Step 5.6

Pull terms out from under the radical.

$y = \pm \frac{3}{4} \sqrt{(x + 7) (- x + 1^{2})}$

Step 5.7

One to any power is one.

$y = \pm \frac{3}{4} \sqrt{(x + 7) (- x + 1)}$

Step 5.8

Combine $\frac{3}{4}$ and $\sqrt{(x + 7) (- x + 1)}$ .

$y = \pm \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

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.

$y = \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

Step 6.2

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

$y = - \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

Step 6.3

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

$y = \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

$y = - \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

$y = \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$

$y = - \frac{3 \sqrt{(x + 7) (- x + 1)}}{4}$