Algebra Examples

$- 2 y^{2} + x - 20 y - 49 = 0$

Step 1

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

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Step 1.1

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

$x - 20 y - 49 = 2 y^{2}$

Step 1.2

Add $20 y$ to both sides of the equation.

$x - 49 = 2 y^{2} + 20 y$

Step 1.3

Add $49$ to both sides of the equation.

$x = 2 y^{2} + 20 y + 49$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $2 y^{2} + 20 y + 49$ .

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Step 2.1.1.1

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

$a = 2$

$b = 20$

$c = 49$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$d = \frac{20}{2 \cdot 2}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $20$ and $2$ .

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Step 2.1.1.3.2.1.1

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot 2}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

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

$d = \frac{2 \cdot 10}{2 (2)}$

Step 2.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 2}$

Step 2.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{10}{2}$

Step 2.1.1.3.2.2

Cancel the common factor of $10$ and $2$ .

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Step 2.1.1.3.2.2.1

Factor $2$ out of $10$ .

$d = \frac{2 \cdot 5}{2}$

Step 2.1.1.3.2.2.2

Cancel the common factors.

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Step 2.1.1.3.2.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 5}{2 (1)}$

Step 2.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Step 2.1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{5}{1}$

Step 2.1.1.3.2.2.2.4

Divide $5$ by $1$ .

$d = 5$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = 49 - \frac{20^{2}}{4 \cdot 2}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Raise $20$ to the power of $2$ .

$e = 49 - \frac{400}{4 \cdot 2}$

Step 2.1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = 49 - \frac{400}{8}$

Step 2.1.1.4.2.1.3

Divide $400$ by $8$ .

$e = 49 - 1 \cdot 50$

Step 2.1.1.4.2.1.4

Multiply $- 1$ by $50$ .

$e = 49 - 50$

Step 2.1.1.4.2.2

Subtract $50$ from $49$ .

$e = - 1$

Step 2.1.1.5

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

$2 {(y + 5)}^{2} - 1$

Step 2.1.2

Set $x$ equal to the new right side.

$x = 2 {(y + 5)}^{2} - 1$

Step 2.2

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

$a = 2$

$h = - 1$

$k = - 5$

Step 2.3

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

Opens Right

Step 2.4

Find the vertex $(h, k)$ .

$(- 1, - 5)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(- \frac{7}{8}, - 5)$

Step 2.7

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

$y = - 5$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

$x = - \frac{9}{8}$

Step 2.9

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

Direction: Opens Right

Vertex: $(- 1, - 5)$

Focus: $(- \frac{7}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{9}{8}$

Direction: Opens Right

Vertex: $(- 1, - 5)$

Focus: $(- \frac{7}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{9}{8}$

Step 3

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 3.1

Substitute the $x$ value $0$ into $f (x) = - \frac{10 + \sqrt{2 (x + 1)}}{2}$ . In this case, the point is $(0, - 5.70710678)$ .

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Step 3.1.1

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

$f (0) = - \frac{10 + \sqrt{2 ((0) + 1)}}{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Add $0$ and $1$ .

$f (0) = - \frac{10 + \sqrt{2 \cdot 1}}{2}$

Step 3.1.2.1.2

Multiply $2$ by $1$ .

$f (0) = - \frac{10 + \sqrt{2}}{2}$

Step 3.1.2.2

The final answer is $- \frac{10 + \sqrt{2}}{2}$ .

$y = - \frac{10 + \sqrt{2}}{2}$

Step 3.1.3

Convert $- \frac{10 + \sqrt{2}}{2}$ to decimal.

$= - 5.70710678$

Step 3.2

Substitute the $x$ value $0$ into $f (x) = - \frac{10 - \sqrt{2 (x + 1)}}{2}$ . In this case, the point is $(0, - 4.29289321)$ .

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Step 3.2.1

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

$f (0) = - \frac{10 - \sqrt{2 ((0) + 1)}}{2}$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify the numerator.

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Step 3.2.2.1.1

Add $0$ and $1$ .

$f (0) = - \frac{10 - \sqrt{2 \cdot 1}}{2}$

Step 3.2.2.1.2

Multiply $2$ by $1$ .

$f (0) = - \frac{10 - \sqrt{2}}{2}$

Step 3.2.2.2

The final answer is $- \frac{10 - \sqrt{2}}{2}$ .

$y = - \frac{10 - \sqrt{2}}{2}$

Step 3.2.3

Convert $- \frac{10 - \sqrt{2}}{2}$ to decimal.

$= - 4.29289321$

Step 3.3

Substitute the $x$ value $1$ into $f (x) = - \frac{10 + \sqrt{2 (x + 1)}}{2}$ . In this case, the point is $(1, - 6)$ .

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Step 3.3.1

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

$f (1) = - \frac{10 + \sqrt{2 ((1) + 1)}}{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the numerator.

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Step 3.3.2.1.1

Add $1$ and $1$ .

$f (1) = - \frac{10 + \sqrt{2 \cdot 2}}{2}$

Step 3.3.2.1.2

Multiply $2$ by $2$ .

$f (1) = - \frac{10 + \sqrt{4}}{2}$

Step 3.3.2.1.3

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

$f (1) = - \frac{10 + \sqrt{2^{2}}}{2}$

Step 3.3.2.1.4

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

$f (1) = - \frac{10 + 2}{2}$

Step 3.3.2.1.5

Add $10$ and $2$ .

$f (1) = - \frac{12}{2}$

Step 3.3.2.2

Simplify the expression.

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Step 3.3.2.2.1

Divide $12$ by $2$ .

$f (1) = - 1 \cdot 6$

Step 3.3.2.2.2

Multiply $- 1$ by $6$ .

$f (1) = - 6$

Step 3.3.2.3

The final answer is $- 6$ .

$y = - 6$

Step 3.3.3

Convert $- 6$ to decimal.

$= - 6$

Step 3.4

Substitute the $x$ value $1$ into $f (x) = - \frac{10 - \sqrt{2 (x + 1)}}{2}$ . In this case, the point is $(1, - 4)$ .

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Step 3.4.1

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

$f (1) = - \frac{10 - \sqrt{2 ((1) + 1)}}{2}$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify the numerator.

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Step 3.4.2.1.1

Add $1$ and $1$ .

$f (1) = - \frac{10 - \sqrt{2 \cdot 2}}{2}$

Step 3.4.2.1.2

Multiply $2$ by $2$ .

$f (1) = - \frac{10 - \sqrt{4}}{2}$

Step 3.4.2.1.3

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

$f (1) = - \frac{10 - \sqrt{2^{2}}}{2}$

Step 3.4.2.1.4

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

$f (1) = - \frac{10 - 1 \cdot 2}{2}$

Step 3.4.2.1.5

Multiply $- 1$ by $2$ .

$f (1) = - \frac{10 - 2}{2}$

Step 3.4.2.1.6

Subtract $2$ from $10$ .

$f (1) = - \frac{8}{2}$

Step 3.4.2.2

Simplify the expression.

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Step 3.4.2.2.1

Divide $8$ by $2$ .

$f (1) = - 1 \cdot 4$

Step 3.4.2.2.2

Multiply $- 1$ by $4$ .

$f (1) = - 4$

Step 3.4.2.3

The final answer is $- 4$ .

$y = - 4$

Step 3.4.3

Convert $- 4$ to decimal.

$= - 4$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 5 \\ 0 & - 5.71 \\ 0 & - 4.29 \\ 1 & - 6 \\ 1 & - 4 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 1, - 5)$

Focus: $(- \frac{7}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{9}{8}$

$\begin{array}{c} x & y \\ - 1 & - 5 \\ 0 & - 5.71 \\ 0 & - 4.29 \\ 1 & - 6 \\ 1 & - 4 \end{array}$

Step 5