Precalculus Examples

$3 x - y^{2} = 4 x + 25$

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

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Step 1.1.1.1

Subtract $4 x$ from both sides of the equation.

$3 x - y^{2} - 4 x = 25$

Step 1.1.1.2

Subtract $4 x$ from $3 x$ .

$- y^{2} - x = 25$

Step 1.1.2

Add $y^{2}$ to both sides of the equation.

$- x = 25 + y^{2}$

Step 1.1.3

Divide each term in $- x = 25 + y^{2}$ by $- 1$ and simplify.

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Step 1.1.3.1

Divide each term in $- x = 25 + y^{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 1.1.3.2.2

Divide $x$ by $1$ .

$x = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Divide $25$ by $- 1$ .

$x = - 25 + \frac{y^{2}}{- 1}$

Step 1.1.3.3.1.2

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

$x = - 25 - 1 \cdot y^{2}$

Step 1.1.3.3.1.3

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$x = - 25 - y^{2}$

Step 1.2

Complete the square for $- 25 - y^{2}$ .

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Step 1.2.1

Reorder $- 25$ and $- y^{2}$ .

$- y^{2} - 25$

Step 1.2.2

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

$a = - 1$

$b = 0$

$c = - 25$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{0}{2 \cdot - 1}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.4.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 1}$

Step 1.2.4.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 1.2.4.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 1.2.4.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - 25 - \frac{0^{2}}{4 \cdot - 1}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = - 25 - \frac{0}{4 \cdot - 1}$

Step 1.2.5.2.1.2

Multiply $4$ by $- 1$ .

$e = - 25 - \frac{0}{- 4}$

Step 1.2.5.2.1.3

Divide $0$ by $- 4$ .

$e = - 25 - 0$

Step 1.2.5.2.1.4

Multiply $- 1$ by $0$ .

$e = - 25 + 0$

Step 1.2.5.2.2

Add $- 25$ and $0$ .

$e = - 25$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y + 0)}^{2} - 25$ .

$- {(y + 0)}^{2} - 25$

Step 1.3

Set $x$ equal to the new right side.

$x = - {(y + 0)}^{2} - 25$

Step 2

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

$a = - 1$

$h = - 25$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(- 25, 0)$

Step 4