Algebra Examples

$x^{2} - 2 x + y - 1 = 0$

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.

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

Step 1.1.2

Add $2 x$ to both sides of the equation.

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

Step 1.1.3

Add $1$ to both sides of the equation.

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

Step 1.2

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

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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 $c$ .

$a = - 1$

$b = 2$

$c = 1$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Rewrite the expression.

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

Step 1.2.3.2.1.3

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

$d = - 1 \cdot 1$

Step 1.2.3.2.2

Multiply $- 1$ by $1$ .

$d = - 1$

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

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

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 $2$ to the power of $2$ .

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

Step 1.2.4.2.1.2

Multiply $4$ by $- 1$ .

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

Step 1.2.4.2.1.3

Divide $4$ by $- 4$ .

$e = 1 - - 1$

Step 1.2.4.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 1 + 1$

Step 1.2.4.2.2

Add $1$ and $1$ .

$e = 2$

Step 1.2.5

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

Find $p$ , the distance from the vertex to the focus.

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Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 5.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{9}{4}$

Step 6