Algebra Examples

${(x + 3)}^{2} = - (y - 2)$

Step 1

Isolate $y$ to the left side of the equation.

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Step 1.1

Rewrite the equation as $- (y - 2) = {(x + 3)}^{2}$ .

$- (y - 2) = {(x + 3)}^{2}$

Step 1.2

Simplify $- (y - 2)$ .

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Step 1.2.1

Apply the distributive property.

$- y - - 2 = {(x + 3)}^{2}$

Step 1.2.2

Multiply $- 1$ by $- 2$ .

$- y + 2 = {(x + 3)}^{2}$

Step 1.3

Subtract $2$ from both sides of the equation.

$- y = {(x + 3)}^{2} - 2$

Step 1.4

Divide each term in $- y = {(x + 3)}^{2} - 2$ by $- 1$ and simplify.

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Step 1.4.1

Divide each term in $- y = {(x + 3)}^{2} - 2$ by $- 1$ .

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

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

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

Step 1.4.2.2

Divide $y$ by $1$ .

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

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Move the negative one from the denominator of $\frac{{(x + 3)}^{2}}{- 1}$ .

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

Step 1.4.3.1.2

Rewrite $- 1 \cdot {(x + 3)}^{2}$ as $- {(x + 3)}^{2}$ .

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

Step 1.4.3.1.3

Divide $- 2$ by $- 1$ .

$y = - {(x + 3)}^{2} + 2$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 1$

$h = - 3$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 3, 2)$

Step 4

Find $p$ , the distance from the vertex to the focus.

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Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 1}$

Step 4.3

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 5.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 3, \frac{7}{4})$

Step 6