Algebra Examples

$3 x - y^{2} = 4 x + 25$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$3 x - y^{2} - 4 x = 25$

Step 1.2

Subtract $4 x$ from $3 x$ .

$- y^{2} - x = 25$

Step 2

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

$- x = 25 + y^{2}$

Step 3

Divide each term in $- x = 25 + y^{2}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- x = 25 + y^{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 3.2.2

Divide $x$ by $1$ .

$x = \frac{25}{- 1} + \frac{y^{2}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Divide $25$ by $- 1$ .

$x = - 25 + \frac{y^{2}}{- 1}$

Step 3.3.1.2

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

$x = - 25 - 1 \cdot y^{2}$

Step 3.3.1.3

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$x = - 25 - y^{2}$

Step 4

Find the properties of the given parabola.

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Step 4.1

Rewrite the equation in vertex form.

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Step 4.1.1

Reorder $- 25$ and $- y^{2}$ .

$x = - y^{2} - 25$

Step 4.1.2

Complete the square for $- y^{2} - 25$ .

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Step 4.1.2.1

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

$a = - 1$

$b = 0$

$c = - 25$

Step 4.1.2.2

Consider the vertex form of a parabola.

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

Step 4.1.2.3

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

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Step 4.1.2.3.1

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

$d = \frac{0}{2 \cdot - 1}$

Step 4.1.2.3.2

Simplify the right side.

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Step 4.1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 4.1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 1}$

Step 4.1.2.3.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 4.1.2.3.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 4.1.2.3.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 4.1.2.4

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

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Step 4.1.2.4.1

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

$e = - 25 - \frac{0^{2}}{4 \cdot - 1}$

Step 4.1.2.4.2

Simplify the right side.

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Step 4.1.2.4.2.1

Simplify each term.

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Step 4.1.2.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = - 25 - \frac{0}{4 \cdot - 1}$

Step 4.1.2.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 25 - \frac{0}{- 4}$

Step 4.1.2.4.2.1.3

Divide $0$ by $- 4$ .

$e = - 25 - 0$

Step 4.1.2.4.2.1.4

Multiply $- 1$ by $0$ .

$e = - 25 + 0$

Step 4.1.2.4.2.2

Add $- 25$ and $0$ .

$e = - 25$

Step 4.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y + 0)}^{2} - 25$ .

$- {(y + 0)}^{2} - 25$

Step 4.1.3

Set $x$ equal to the new right side.

$x = - {(y + 0)}^{2} - 25$

Step 4.2

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

$a = - 1$

$h = - 25$

$k = 0$

Step 4.3

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Step 4.4

Find the vertex $(h, k)$ .

$(- 25, 0)$

Step 4.5

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

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Step 4.5.1

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

$\frac{1}{4 a}$

Step 4.5.2

Substitute the value of $a$ into the formula.

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

Step 4.5.3

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

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Step 4.5.3.1

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

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

Step 4.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 4.6

Find the focus.

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Step 4.6.1

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 4.6.2

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

$(- \frac{101}{4}, 0)$

Step 4.7

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

$y = 0$

Step 4.8

Find the directrix.

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Step 4.8.1

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 4.8.2

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

$x = - \frac{99}{4}$

Step 4.9

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

Direction: Opens Left

Vertex: $(- 25, 0)$

Focus: $(- \frac{101}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{99}{4}$

Direction: Opens Left

Vertex: $(- 25, 0)$

Focus: $(- \frac{101}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{99}{4}$

Step 5

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 5.1

Substitute the $x$ value $- 27$ into $f (x) = \sqrt{- x - 25}$ . In this case, the point is $(- 27, 1.41421356)$ .

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Step 5.1.1

Replace the variable $x$ with $- 27$ in the expression.

$f (- 27) = \sqrt{- (- 27) - 25}$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Multiply $- 1$ by $- 27$ .

$f (- 27) = \sqrt{27 - 25}$

Step 5.1.2.2

Subtract $25$ from $27$ .

$f (- 27) = \sqrt{2}$

Step 5.1.2.3

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

$y = \sqrt{2}$

Step 5.1.3

Convert $\sqrt{2}$ to decimal.

$= 1.41421356$

Step 5.2

Substitute the $x$ value $- 27$ into $f (x) = - \sqrt{- x - 25}$ . In this case, the point is $(- 27, - 1.41421356)$ .

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Step 5.2.1

Replace the variable $x$ with $- 27$ in the expression.

$f (- 27) = - \sqrt{- (- 27) - 25}$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Multiply $- 1$ by $- 27$ .

$f (- 27) = - \sqrt{27 - 25}$

Step 5.2.2.2

Subtract $25$ from $27$ .

$f (- 27) = - \sqrt{2}$

Step 5.2.2.3

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

$y = - \sqrt{2}$

Step 5.2.3

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

$= - 1.41421356$

Step 5.3

Substitute the $x$ value $- 26$ into $f (x) = \sqrt{- x - 25}$ . In this case, the point is $(- 26, 1)$ .

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Step 5.3.1

Replace the variable $x$ with $- 26$ in the expression.

$f (- 26) = \sqrt{- (- 26) - 25}$

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Multiply $- 1$ by $- 26$ .

$f (- 26) = \sqrt{26 - 25}$

Step 5.3.2.2

Subtract $25$ from $26$ .

$f (- 26) = \sqrt{1}$

Step 5.3.2.3

Any root of $1$ is $1$ .

$f (- 26) = 1$

Step 5.3.2.4

The final answer is $1$ .

$y = 1$

Step 5.3.3

Convert $1$ to decimal.

$= 1$

Step 5.4

Substitute the $x$ value $- 26$ into $f (x) = - \sqrt{- x - 25}$ . In this case, the point is $(- 26, - 1)$ .

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Step 5.4.1

Replace the variable $x$ with $- 26$ in the expression.

$f (- 26) = - \sqrt{- (- 26) - 25}$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Multiply $- 1$ by $- 26$ .

$f (- 26) = - \sqrt{26 - 25}$

Step 5.4.2.2

Subtract $25$ from $26$ .

$f (- 26) = - \sqrt{1}$

Step 5.4.2.3

Any root of $1$ is $1$ .

$f (- 26) = - 1 \cdot 1$

Step 5.4.2.4

Multiply $- 1$ by $1$ .

$f (- 26) = - 1$

Step 5.4.2.5

The final answer is $- 1$ .

$y = - 1$

Step 5.4.3

Convert $- 1$ to decimal.

$= - 1$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 27 & 1.41 \\ - 27 & - 1.41 \\ - 26 & 1 \\ - 26 & - 1 \\ - 25 & 0 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 25, 0)$

Focus: $(- \frac{101}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{99}{4}$

$\begin{array}{c} x & y \\ - 27 & 1.41 \\ - 27 & - 1.41 \\ - 26 & 1 \\ - 26 & - 1 \\ - 25 & 0 \end{array}$

Step 7