Algebra Examples

$y = 9 - 8 x + 2 x^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify $9 - 8 x + 2 x^{2}$ .

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Step 1.1.1

Move $9$ .

$y = - 8 x + 2 x^{2} + 9$

Step 1.1.2

Reorder $- 8 x$ and $2 x^{2}$ .

$y = 2 x^{2} - 8 x + 9$

Step 1.2

Complete the square for $2 x^{2} - 8 x + 9$ .

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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 $c$ .

$a = 2$

$b = - 8$

$c = 9$

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

$d = \frac{- 8}{2 \cdot 2}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 2}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 2$ .

$d = \frac{2 \cdot - 4}{2 (2)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 2}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

Cancel the common factor of $- 4$ and $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $- 4$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 9 - \frac{{(- 8)}^{2}}{4 \cdot 2}$

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 $- 8$ to the power of $2$ .

$e = 9 - \frac{64}{4 \cdot 2}$

Step 1.2.4.2.1.2

Multiply $4$ by $2$ .

$e = 9 - \frac{64}{8}$

Step 1.2.4.2.1.3

Divide $64$ by $8$ .

$e = 9 - 1 \cdot 8$

Step 1.2.4.2.1.4

Multiply $- 1$ by $8$ .

$e = 9 - 8$

Step 1.2.4.2.2

Subtract $8$ from $9$ .

$e = 1$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = 2$

$h = 2$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(2, 1)$

Step 4