Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.2.2

Divide $y$ by $1$ .

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

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

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

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

Step 1.1.2.3.1.2

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

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

Step 1.1.2.3.1.3

Multiply $2$ by $- 1$ .

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

Step 1.1.2.3.1.4

Divide $1$ by $- 1$ .

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

Step 1.1.2.3.1.5

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.6

Divide $x^{2}$ by $1$ .

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

Step 1.2

Complete the square for $- 2 x - 1 + x^{2}$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Move $- 1$ .

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

Step 1.2.1.2

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

$x^{2} - 2 x - 1$

Step 1.2.2

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

$a = 1$

$b = - 2$

$c = - 1$

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

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

Factor $2$ out of $- 2$ .

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

Step 1.2.4.2.2

Cancel the common factors.

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Step 1.2.4.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 1.2.4.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.2.3

Rewrite the expression.

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

Step 1.2.4.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

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

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

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

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

Step 1.2.5.2.1.2

Multiply $4$ by $1$ .

$e = - 1 - \frac{4}{4}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.3.1

Cancel the common factor.

$e = - 1 - \frac{4}{4}$

Step 1.2.5.2.1.3.2

Rewrite the expression.

$e = - 1 - 1 \cdot 1$

Step 1.2.5.2.1.4

Multiply $- 1$ by $1$ .

$e = - 1 - 1$

Step 1.2.5.2.2

Subtract $1$ from $- 1$ .

$e = - 2$

Step 1.2.6

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = 1$

$h = 1$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(1, - 2)$

Step 4