Algebra Examples

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

Step 1

Add $y^{2}$ to both sides of the equation.

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

Step 2

Divide each term in $3 x = 8 y + 1 + y^{2}$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x = 8 y + 1 + y^{2}$ by $3$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $x$ by $1$ .

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

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

Simplify $\frac{8 y}{3} + \frac{1}{3} + \frac{y^{2}}{3}$ .

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Step 3.1.1.1

Move $\frac{1}{3}$ .

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

Step 3.1.1.2

Reorder $\frac{8 y}{3}$ and $\frac{y^{2}}{3}$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

$a = \frac{1}{3}$

$b = \frac{8}{3}$

$c = \frac{1}{3}$

Step 3.1.2.2

Consider the vertex form of a parabola.

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

Step 3.1.2.3

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

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Step 3.1.2.3.1

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

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

Step 3.1.2.3.2

Simplify the right side.

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Step 3.1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2.3.2.2

Cancel the common factor of $2$ .

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Step 3.1.2.3.2.2.1

Factor $2$ out of $8$ .

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

Step 3.1.2.3.2.2.2

Cancel the common factor.

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

Step 3.1.2.3.2.2.3

Rewrite the expression.

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

Step 3.1.2.3.2.3

Multiply $\frac{4}{3}$ by $\frac{1}{\frac{1}{3}}$ .

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

Step 3.1.2.3.2.4

Combine $3$ and $\frac{1}{3}$ .

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

Step 3.1.2.3.2.5

Cancel the common factor of $3$ .

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Step 3.1.2.3.2.5.1

Cancel the common factor.

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

Step 3.1.2.3.2.5.2

Rewrite the expression.

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

Step 3.1.2.3.2.6

Divide $4$ by $1$ .

$d = 4$

Step 3.1.2.4

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

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Step 3.1.2.4.1

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

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

Step 3.1.2.4.2

Simplify the right side.

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Step 3.1.2.4.2.1

Simplify each term.

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Step 3.1.2.4.2.1.1

Simplify the numerator.

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Step 3.1.2.4.2.1.1.1

Apply the product rule to $\frac{8}{3}$ .

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

Step 3.1.2.4.2.1.1.2

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

$e = \frac{1}{3} - \frac{\frac{64}{3^{2}}}{4 (\frac{1}{3})}$

Step 3.1.2.4.2.1.1.3

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

$e = \frac{1}{3} - \frac{\frac{64}{9}}{4 (\frac{1}{3})}$

Step 3.1.2.4.2.1.2

Combine $4$ and $\frac{1}{3}$ .

$e = \frac{1}{3} - \frac{\frac{64}{9}}{\frac{4}{3}}$

Step 3.1.2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{3} - (\frac{64}{9} \cdot \frac{3}{4})$

Step 3.1.2.4.2.1.4

Cancel the common factor of $4$ .

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Step 3.1.2.4.2.1.4.1

Factor $4$ out of $64$ .

$e = \frac{1}{3} - (\frac{4 (16)}{9} \cdot \frac{3}{4})$

Step 3.1.2.4.2.1.4.2

Cancel the common factor.

$e = \frac{1}{3} - (\frac{4 \cdot 16}{9} \cdot \frac{3}{4})$

Step 3.1.2.4.2.1.4.3

Rewrite the expression.

$e = \frac{1}{3} - (\frac{16}{9} \cdot 3)$

Step 3.1.2.4.2.1.5

Cancel the common factor of $3$ .

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Step 3.1.2.4.2.1.5.1

Factor $3$ out of $9$ .

$e = \frac{1}{3} - (\frac{16}{3 (3)} \cdot 3)$

Step 3.1.2.4.2.1.5.2

Cancel the common factor.

$e = \frac{1}{3} - (\frac{16}{3 \cdot 3} \cdot 3)$

Step 3.1.2.4.2.1.5.3

Rewrite the expression.

$e = \frac{1}{3} - \frac{16}{3}$

Step 3.1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 - 16}{3}$

Step 3.1.2.4.2.3

Subtract $16$ from $1$ .

$e = \frac{- 15}{3}$

Step 3.1.2.4.2.4

Divide $- 15$ by $3$ .

$e = - 5$

Step 3.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{3} {(y + 4)}^{2} - 5$ .

$\frac{1}{3} {(y + 4)}^{2} - 5$

Step 3.1.3

Set $x$ equal to the new right side.

$x = \frac{1}{3} \cdot {(y + 4)}^{2} - 5$

Step 3.2

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

$a = \frac{1}{3}$

$h = - 5$

$k = - 4$

Step 3.3

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

Opens Right

Step 3.4

Find the vertex $(h, k)$ .

$(- 5, - 4)$

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.

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

Step 3.5.3

Simplify.

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Step 3.5.3.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{1}{\frac{4}{3}}$

Step 3.5.3.2

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{4})$

Step 3.5.3.3

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

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.

$(- \frac{17}{4}, - 4)$

Step 3.7

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

$y = - 4$

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.

$x = - \frac{23}{4}$

Step 3.9

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

Direction: Opens Right

Vertex: $(- 5, - 4)$

Focus: $(- \frac{17}{4}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = - \frac{23}{4}$

Direction: Opens Right

Vertex: $(- 5, - 4)$

Focus: $(- \frac{17}{4}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = - \frac{23}{4}$

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 $- 4$ into $f (x) = - 4 - \sqrt{3 (x + 5)}$ . In this case, the point is $(- 4, - 5.7320508)$ .

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Step 4.1.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 4 - \sqrt{3 ((- 4) + 5)}$

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 $- 4$ and $5$ .

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

Step 4.1.2.1.2

Multiply $3$ by $1$ .

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

Step 4.1.2.2

The final answer is $- 4 - \sqrt{3}$ .

$y = - 4 - \sqrt{3}$

Step 4.1.3

Convert $- 4 - \sqrt{3}$ to decimal.

$= - 5.7320508$

Step 4.2

Substitute the $x$ value $- 4$ into $f (x) = - 4 + \sqrt{3 (x + 5)}$ . In this case, the point is $(- 4, - 2.26794919)$ .

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Step 4.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 4 + \sqrt{3 ((- 4) + 5)}$

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 $- 4$ and $5$ .

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

Step 4.2.2.1.2

Multiply $3$ by $1$ .

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

Step 4.2.2.2

The final answer is $- 4 + \sqrt{3}$ .

$y = - 4 + \sqrt{3}$

Step 4.2.3

Convert $- 4 + \sqrt{3}$ to decimal.

$= - 2.26794919$

Step 4.3

Substitute the $x$ value $- 3$ into $f (x) = - 4 - \sqrt{3 (x + 5)}$ . In this case, the point is $(- 3, - 6.44948974)$ .

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Step 4.3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 4 - \sqrt{3 ((- 3) + 5)}$

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 $- 3$ and $5$ .

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

Step 4.3.2.1.2

Multiply $3$ by $2$ .

$f (- 3) = - 4 - \sqrt{6}$

Step 4.3.2.2

The final answer is $- 4 - \sqrt{6}$ .

$y = - 4 - \sqrt{6}$

Step 4.3.3

Convert $- 4 - \sqrt{6}$ to decimal.

$= - 6.44948974$

Step 4.4

Substitute the $x$ value $- 3$ into $f (x) = - 4 + \sqrt{3 (x + 5)}$ . In this case, the point is $(- 3, - 1.55051025)$ .

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Step 4.4.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 4 + \sqrt{3 ((- 3) + 5)}$

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 $- 3$ and $5$ .

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

Step 4.4.2.1.2

Multiply $3$ by $2$ .

$f (- 3) = - 4 + \sqrt{6}$

Step 4.4.2.2

The final answer is $- 4 + \sqrt{6}$ .

$y = - 4 + \sqrt{6}$

Step 4.4.3

Convert $- 4 + \sqrt{6}$ to decimal.

$= - 1.55051025$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & - 4 \\ - 4 & - 5.73 \\ - 4 & - 2.27 \\ - 3 & - 6.45 \\ - 3 & - 1.55 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 5, - 4)$

Focus: $(- \frac{17}{4}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = - \frac{23}{4}$

$\begin{array}{c} x & y \\ - 5 & - 4 \\ - 4 & - 5.73 \\ - 4 & - 2.27 \\ - 3 & - 6.45 \\ - 3 & - 1.55 \end{array}$

Step 6