Precalculus Examples

$x^{2} - 6 x + y + 8 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

$- 6 x + y + 8 = - x^{2}$

Step 1.1.2

Add $6 x$ to both sides of the equation.

$y + 8 = - x^{2} + 6 x$

Step 1.1.3

Subtract $8$ from both sides of the equation.

$y = - x^{2} + 6 x - 8$

Step 1.2

Complete the square for $- x^{2} + 6 x - 8$ .

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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 $c$ .

$a = - 1$

$b = 6$

$c = - 8$

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 $d = \frac{b}{2 a}$ .

$d = \frac{6}{2 \cdot - 1}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $6$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot - 1}$

Step 1.2.3.2.1.2

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$d = - 1 \cdot 3$

Step 1.2.3.2.2

Multiply $- 1$ by $3$ .

$d = - 3$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 8 - \frac{6^{2}}{4 \cdot - 1}$

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 $6$ to the power of $2$ .

$e = - 8 - \frac{36}{4 \cdot - 1}$

Step 1.2.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 8 - \frac{36}{- 4}$

Step 1.2.4.2.1.3

Divide $36$ by $- 4$ .

$e = - 8 - - 9$

Step 1.2.4.2.1.4

Multiply $- 1$ by $- 9$ .

$e = - 8 + 9$

Step 1.2.4.2.2

Add $- 8$ and $9$ .

$e = 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 3)}^{2} + 1$ .

$- {(x - 3)}^{2} + 1$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 1$

$h = 3$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(3, 1)$

Step 4