Precalculus Examples

$- \frac{1}{3} (x - 3) = {(y + 5)}^{2}$

Step 1

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

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Step 1.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{1}{3} \cdot (x - 3)) = - 3 {(y + 5)}^{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $- 3 (- \frac{1}{3} \cdot (x - 3))$ .

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Step 1.2.1.1

Apply the distributive property.

$- 3 (- \frac{1}{3} x - \frac{1}{3} \cdot - 3) = - 3 {(y + 5)}^{2}$

Step 1.2.1.2

Combine $x$ and $\frac{1}{3}$ .

$- 3 (- \frac{x}{3} - \frac{1}{3} \cdot - 3) = - 3 {(y + 5)}^{2}$

Step 1.2.1.3

Cancel the common factor of $3$ .

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Step 1.2.1.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$- 3 (- \frac{x}{3} + \frac{- 1}{3} \cdot - 3) = - 3 {(y + 5)}^{2}$

Step 1.2.1.3.2

Factor $3$ out of $- 3$ .

$- 3 (- \frac{x}{3} + \frac{- 1}{3} \cdot (3 (- 1))) = - 3 {(y + 5)}^{2}$

Step 1.2.1.3.3

Cancel the common factor.

$- 3 (- \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot - 1)) = - 3 {(y + 5)}^{2}$

Step 1.2.1.3.4

Rewrite the expression.

$- 3 (- \frac{x}{3} - 1 \cdot - 1) = - 3 {(y + 5)}^{2}$

Step 1.2.1.4

Multiply $- 1$ by $- 1$ .

$- 3 (- \frac{x}{3} + 1) = - 3 {(y + 5)}^{2}$

Step 1.2.1.5

Apply the distributive property.

$- 3 (- \frac{x}{3}) - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.6

Cancel the common factor of $3$ .

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Step 1.2.1.6.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$- 3 \frac{- x}{3} - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.6.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- x}{3} - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.6.3

Cancel the common factor.

$3 \cdot - 1 \frac{- x}{3} - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.6.4

Rewrite the expression.

$- - x - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.7

Multiply.

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Step 1.2.1.7.1

Multiply $- 1$ by $- 1$ .

$1 x - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.7.2

Simplify.

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Step 1.2.1.7.2.1

Multiply $x$ by $1$ .

$x - 3 \cdot 1 = - 3 {(y + 5)}^{2}$

Step 1.2.1.7.2.2

Multiply $- 3$ by $1$ .

$x - 3 = - 3 {(y + 5)}^{2}$

Step 1.3

Add $3$ to both sides of the equation.

$x = - 3 {(y + 5)}^{2} + 3$

Step 2

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

$a = - 3$

$h = 3$

$k = - 5$

Step 3

Find the vertex $(h, k)$ .

$(3, - 5)$

Step 4