Pre-Algebra Examples

$y^{2} + 6 y - 8 x + 1 = 0$

Step 1

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

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Step 1.1

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

$6 y - 8 x + 1 = - y^{2}$

Step 1.2

Subtract $6 y$ from both sides of the equation.

$- 8 x + 1 = - y^{2} - 6 y$

Step 1.3

Subtract $1$ from both sides of the equation.

$- 8 x = - y^{2} - 6 y - 1$

Step 2

Divide each term in $- 8 x = - y^{2} - 6 y - 1$ by $- 8$ and simplify.

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Step 2.1

Divide each term in $- 8 x = - y^{2} - 6 y - 1$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- y^{2}}{- 8} + \frac{- 6 y}{- 8} + \frac{- 1}{- 8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- y^{2}}{- 8} + \frac{- 6 y}{- 8} + \frac{- 1}{- 8}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 8} + \frac{- 6 y}{- 8} + \frac{- 1}{- 8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{8} + \frac{- 6 y}{- 8} + \frac{- 1}{- 8}$

Step 2.3.1.2

Cancel the common factor of $- 6$ and $- 8$ .

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Step 2.3.1.2.1

Factor $- 2$ out of $- 6 y$ .

$x = \frac{y^{2}}{8} + \frac{- 2 (3 y)}{- 8} + \frac{- 1}{- 8}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $- 2$ out of $- 8$ .

$x = \frac{y^{2}}{8} + \frac{- 2 (3 y)}{- 2 (4)} + \frac{- 1}{- 8}$

Step 2.3.1.2.2.2

Cancel the common factor.

$x = \frac{y^{2}}{8} + \frac{- 2 (3 y)}{- 2 \cdot 4} + \frac{- 1}{- 8}$

Step 2.3.1.2.2.3

Rewrite the expression.

$x = \frac{y^{2}}{8} + \frac{3 y}{4} + \frac{- 1}{- 8}$

Step 2.3.1.3

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{8} + \frac{3 y}{4} + \frac{1}{8}$

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

Complete the square for $\frac{y^{2}}{8} + \frac{3 y}{4} + \frac{1}{8}$ .

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Step 3.1.1.1

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

$a = \frac{1}{8}$

$b = \frac{3}{4}$

$c = \frac{1}{8}$

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

$d = \frac{\frac{3}{4}}{2 (\frac{1}{8})}$

Step 3.1.1.3.2

Simplify the right side.

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Step 3.1.1.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} \cdot \frac{1}{2 (\frac{1}{8})}$

Step 3.1.1.3.2.2

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2}{8}}$

Step 3.1.1.3.2.3

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

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Step 3.1.1.3.2.3.1

Factor $2$ out of $2$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 (1)}{8}}$

Step 3.1.1.3.2.3.2

Cancel the common factors.

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Step 3.1.1.3.2.3.2.1

Factor $2$ out of $8$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 3.1.1.3.2.3.2.2

Cancel the common factor.

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 3.1.1.3.2.3.2.3

Rewrite the expression.

$d = \frac{3}{4} \cdot \frac{1}{\frac{1}{4}}$

Step 3.1.1.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} (1 \cdot 4)$

Step 3.1.1.3.2.5

Cancel the common factor of $4$ .

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Step 3.1.1.3.2.5.1

Factor $4$ out of $1 \cdot 4$ .

$d = \frac{3}{4} (4 \cdot 1)$

Step 3.1.1.3.2.5.2

Cancel the common factor.

$d = \frac{3}{4} (4 \cdot 1)$

Step 3.1.1.3.2.5.3

Rewrite the expression.

$d = 3$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

$e = \frac{1}{8} - \frac{{(\frac{3}{4})}^{2}}{4 (\frac{1}{8})}$

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

Simplify the numerator.

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Step 3.1.1.4.2.1.1.1

Apply the product rule to $\frac{3}{4}$ .

$e = \frac{1}{8} - \frac{\frac{3^{2}}{4^{2}}}{4 (\frac{1}{8})}$

Step 3.1.1.4.2.1.1.2

Raise $3$ to the power of $2$ .

$e = \frac{1}{8} - \frac{\frac{9}{4^{2}}}{4 (\frac{1}{8})}$

Step 3.1.1.4.2.1.1.3

Raise $4$ to the power of $2$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{4 (\frac{1}{8})}$

Step 3.1.1.4.2.1.2

Combine $4$ and $\frac{1}{8}$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4}{8}}$

Step 3.1.1.4.2.1.3

Cancel the common factor of $4$ and $8$ .

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Step 3.1.1.4.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 (1)}{8}}$

Step 3.1.1.4.2.1.3.2

Cancel the common factors.

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Step 3.1.1.4.2.1.3.2.1

Factor $4$ out of $8$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{1}{2}}$

Step 3.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{8} - (\frac{9}{16} \cdot 2)$

Step 3.1.1.4.2.1.5

Cancel the common factor of $2$ .

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Step 3.1.1.4.2.1.5.1

Factor $2$ out of $16$ .

$e = \frac{1}{8} - (\frac{9}{2 (8)} \cdot 2)$

Step 3.1.1.4.2.1.5.2

Cancel the common factor.

$e = \frac{1}{8} - (\frac{9}{2 \cdot 8} \cdot 2)$

Step 3.1.1.4.2.1.5.3

Rewrite the expression.

$e = \frac{1}{8} - \frac{9}{8}$

Step 3.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 - 9}{8}$

Step 3.1.1.4.2.3

Subtract $9$ from $1$ .

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

Step 3.1.1.4.2.4

Divide $- 8$ by $8$ .

$e = - 1$

Step 3.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(y + 3)}^{2} - 1$ .

$\frac{1}{8} {(y + 3)}^{2} - 1$

Step 3.1.2

Set $x$ equal to the new right side.

$x = \frac{1}{8} \cdot {(y + 3)}^{2} - 1$

Step 3.2

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

$a = \frac{1}{8}$

$h = - 1$

$k = - 3$

Step 3.3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 3.4

Find the vertex $(h, k)$ .

$(- 1, - 3)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{8}}$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Step 3.5.3.2

Cancel the common factor of $4$ and $8$ .

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Step 3.5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Step 3.5.3.2.2

Cancel the common factors.

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Step 3.5.3.2.2.1

Factor $4$ out of $8$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.5.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{2}}$

Step 3.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 3.5.3.4

Multiply $2$ by $1$ .

$2$

Step 3.6

Find the focus.

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Step 3.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 3.6.2

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

$(1, - 3)$

Step 3.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = - 3$

Step 3.8

Find the directrix.

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Step 3.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 3.8.2

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

$x = - 3$

Step 3.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Right

Vertex: $(- 1, - 3)$

Focus: $(1, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = - 3$

Direction: Opens Right

Vertex: $(- 1, - 3)$

Focus: $(1, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = - 3$

Step 4

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 4.1

Substitute the $x$ value $0$ into $f (x) = - 3 + 2 \sqrt{2 (x + 1)}$ . In this case, the point is $(0, - 0.17157287)$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 + 2 \sqrt{2 ((0) + 1)}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Add $0$ and $1$ .

$f (0) = - 3 + 2 \sqrt{2 \cdot 1}$

Step 4.1.2.1.2

Multiply $2$ by $1$ .

$f (0) = - 3 + 2 \sqrt{2}$

Step 4.1.2.2

The final answer is $- 3 + 2 \sqrt{2}$ .

$y = - 3 + 2 \sqrt{2}$

Step 4.1.3

Convert $- 3 + 2 \sqrt{2}$ to decimal.

$= - 0.17157287$

Step 4.2

Substitute the $x$ value $0$ into $f (x) = - 3 - 2 \sqrt{2 (x + 1)}$ . In this case, the point is $(0, - 5.82842712)$ .

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Step 4.2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 - 2 \sqrt{2 ((0) + 1)}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Add $0$ and $1$ .

$f (0) = - 3 - 2 \sqrt{2 \cdot 1}$

Step 4.2.2.1.2

Multiply $2$ by $1$ .

$f (0) = - 3 - 2 \sqrt{2}$

Step 4.2.2.2

The final answer is $- 3 - 2 \sqrt{2}$ .

$y = - 3 - 2 \sqrt{2}$

Step 4.2.3

Convert $- 3 - 2 \sqrt{2}$ to decimal.

$= - 5.82842712$

Step 4.3

Substitute the $x$ value $1$ into $f (x) = - 3 + 2 \sqrt{2 (x + 1)}$ . In this case, the point is $(1, 1)$ .

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Step 4.3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - 3 + 2 \sqrt{2 ((1) + 1)}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Add $1$ and $1$ .

$f (1) = - 3 + 2 \sqrt{2 \cdot 2}$

Step 4.3.2.1.2

Multiply $2$ by $2$ .

$f (1) = - 3 + 2 \sqrt{4}$

Step 4.3.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (1) = - 3 + 2 \sqrt{2^{2}}$

Step 4.3.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (1) = - 3 + 2 \cdot 2$

Step 4.3.2.1.5

Multiply $2$ by $2$ .

$f (1) = - 3 + 4$

Step 4.3.2.2

Add $- 3$ and $4$ .

$f (1) = 1$

Step 4.3.2.3

The final answer is $1$ .

$y = 1$

Step 4.3.3

Convert $1$ to decimal.

$= 1$

Step 4.4

Substitute the $x$ value $1$ into $f (x) = - 3 - 2 \sqrt{2 (x + 1)}$ . In this case, the point is $(1, - 7)$ .

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Step 4.4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - 3 - 2 \sqrt{2 ((1) + 1)}$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Add $1$ and $1$ .

$f (1) = - 3 - 2 \sqrt{2 \cdot 2}$

Step 4.4.2.1.2

Multiply $2$ by $2$ .

$f (1) = - 3 - 2 \sqrt{4}$

Step 4.4.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (1) = - 3 - 2 \sqrt{2^{2}}$

Step 4.4.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (1) = - 3 - 2 \cdot 2$

Step 4.4.2.1.5

Multiply $- 2$ by $2$ .

$f (1) = - 3 - 4$

Step 4.4.2.2

Subtract $4$ from $- 3$ .

$f (1) = - 7$

Step 4.4.2.3

The final answer is $- 7$ .

$y = - 7$

Step 4.4.3

Convert $- 7$ to decimal.

$= - 7$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 3 \\ 0 & - 0.17 \\ 0 & - 5.83 \\ 1 & 1 \\ 1 & - 7 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 1, - 3)$

Focus: $(1, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = - 3$

$\begin{array}{c} x & y \\ - 1 & - 3 \\ 0 & - 0.17 \\ 0 & - 5.83 \\ 1 & 1 \\ 1 & - 7 \end{array}$

Step 6