Precalculus Examples

$y^{2} - 2 x - 10 y + 21 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$- 2 x - 10 y + 21 = - y^{2}$

Step 1.1.1.2

Add $10 y$ to both sides of the equation.

$- 2 x + 21 = - y^{2} + 10 y$

Step 1.1.1.3

Subtract $21$ from both sides of the equation.

$- 2 x = - y^{2} + 10 y - 21$

Step 1.1.2

Divide each term in $- 2 x = - y^{2} + 10 y - 21$ by $- 2$ and simplify.

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Step 1.1.2.1

Divide each term in $- 2 x = - y^{2} + 10 y - 21$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- y^{2}}{- 2} + \frac{10 y}{- 2} + \frac{- 21}{- 2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- y^{2}}{- 2} + \frac{10 y}{- 2} + \frac{- 21}{- 2}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 2} + \frac{10 y}{- 2} + \frac{- 21}{- 2}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{2} + \frac{10 y}{- 2} + \frac{- 21}{- 2}$

Step 1.1.2.3.1.2

Cancel the common factor of $10$ and $- 2$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $10 y$ .

$x = \frac{y^{2}}{2} + \frac{2 (5 y)}{- 2} + \frac{- 21}{- 2}$

Step 1.1.2.3.1.2.2

Move the negative one from the denominator of $\frac{5 y}{- 1}$ .

$x = \frac{y^{2}}{2} - 1 \cdot (5 y) + \frac{- 21}{- 2}$

Step 1.1.2.3.1.3

Rewrite $- 1 \cdot (5 y)$ as $- (5 y)$ .

$x = \frac{y^{2}}{2} - (5 y) + \frac{- 21}{- 2}$

Step 1.1.2.3.1.4

Multiply $5$ by $- 1$ .

$x = \frac{y^{2}}{2} - 5 y + \frac{- 21}{- 2}$

Step 1.1.2.3.1.5

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{2} - 5 y + \frac{21}{2}$

Step 1.2

Complete the square for $\frac{y^{2}}{2} - 5 y + \frac{21}{2}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{1}{2}$

$b = - 5$

$c = \frac{21}{2}$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 5}{2 (\frac{1}{2})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Combine $2$ and $\frac{1}{2}$ .

$d = \frac{- 5}{\frac{2}{2}}$

Step 1.2.3.2.2

Divide $2$ by $2$ .

$d = \frac{- 5}{1}$

Step 1.2.3.2.3

Divide $- 5$ by $1$ .

$d = - 5$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{21}{2} - \frac{{(- 5)}^{2}}{4 (\frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 5$ to the power of $2$ .

$e = \frac{21}{2} - \frac{25}{4 (\frac{1}{2})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{2}$ .

$e = \frac{21}{2} - \frac{25}{\frac{4}{2}}$

Step 1.2.4.2.1.3

Divide $4$ by $2$ .

$e = \frac{21}{2} - \frac{25}{2}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{21 - 25}{2}$

Step 1.2.4.2.3

Subtract $25$ from $21$ .

$e = \frac{- 4}{2}$

Step 1.2.4.2.4

Divide $- 4$ by $2$ .

$e = - 2$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(y - 5)}^{2} - 2$ .

$\frac{1}{2} {(y - 5)}^{2} - 2$

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = \frac{1}{2}$

$h = - 2$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 5)$

Step 4