Algebra Examples

$x^{2} - 6 x + 5 y = - 34$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

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

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Step 1.1.1.1

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

$- 6 x + 5 y = - 34 - x^{2}$

Step 1.1.1.2

Add $6 x$ to both sides of the equation.

$5 y = - 34 - x^{2} + 6 x$

Step 1.1.2

Divide each term in $5 y = - 34 - x^{2} + 6 x$ by $5$ and simplify.

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Step 1.1.2.1

Divide each term in $5 y = - 34 - x^{2} + 6 x$ by $5$ .

$\frac{5 y}{5} = \frac{- 34}{5} + \frac{- x^{2}}{5} + \frac{6 x}{5}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $5$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 34}{5} + \frac{- x^{2}}{5} + \frac{6 x}{5}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 34}{5} + \frac{- x^{2}}{5} + \frac{6 x}{5}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{34}{5} + \frac{- x^{2}}{5} + \frac{6 x}{5}$

Step 1.1.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{34}{5} - \frac{x^{2}}{5} + \frac{6 x}{5}$

Step 1.2

Complete the square for $- \frac{34}{5} - \frac{x^{2}}{5} + \frac{6 x}{5}$ .

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Step 1.2.1

Move $- \frac{34}{5}$ .

$- \frac{x^{2}}{5} + \frac{6 x}{5} - \frac{34}{5}$

Step 1.2.2

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

$a = - \frac{1}{5}$

$b = \frac{6}{5}$

$c = - \frac{34}{5}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{\frac{6}{5}}{2 (- \frac{1}{5})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{6}{5} \cdot \frac{1}{2 (- \frac{1}{5})}$

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Factor $2$ out of $6$ .

$d = \frac{2 (3)}{5} \cdot \frac{1}{2 (- \frac{1}{5})}$

Step 1.2.4.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{5} \cdot \frac{1}{2 (- \frac{1}{5})}$

Step 1.2.4.2.2.3

Rewrite the expression.

$d = \frac{3}{5} \cdot \frac{1}{- \frac{1}{5}}$

Step 1.2.4.2.3

Multiply $\frac{3}{5}$ by $\frac{1}{- \frac{1}{5}}$ .

$d = \frac{3}{5 (- \frac{1}{5})}$

Step 1.2.4.2.4

Multiply $- 1$ by $5$ .

$d = \frac{3}{- 5 (\frac{1}{5})}$

Step 1.2.4.2.5

Combine $- 5$ and $\frac{1}{5}$ .

$d = \frac{3}{\frac{- 5}{5}}$

Step 1.2.4.2.6

Cancel the common factor of $- 5$ and $5$ .

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Step 1.2.4.2.6.1

Factor $5$ out of $- 5$ .

$d = \frac{3}{\frac{5 \cdot - 1}{5}}$

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $5$ out of $5$ .

$d = \frac{3}{\frac{5 \cdot - 1}{5 (1)}}$

Step 1.2.4.2.6.2.2

Cancel the common factor.

$d = \frac{3}{\frac{5 \cdot - 1}{5 \cdot 1}}$

Step 1.2.4.2.6.2.3

Rewrite the expression.

$d = \frac{3}{\frac{- 1}{1}}$

Step 1.2.4.2.6.2.4

Divide $- 1$ by $1$ .

$d = \frac{3}{- 1}$

Step 1.2.4.2.7

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$d = - 1 \cdot 3$

Step 1.2.4.2.8

Multiply $- 1$ by $3$ .

$d = - 3$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{34}{5} - \frac{{(\frac{6}{5})}^{2}}{4 (- \frac{1}{5})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $\frac{6}{5}$ .

$e = - \frac{34}{5} - \frac{\frac{6^{2}}{5^{2}}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.1.2

Raise $6$ to the power of $2$ .

$e = - \frac{34}{5} - \frac{\frac{36}{5^{2}}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.1.3

Raise $5$ to the power of $2$ .

$e = - \frac{34}{5} - \frac{\frac{36}{25}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.2

Simplify the denominator.

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{34}{5} - \frac{\frac{36}{25}}{- 4 (\frac{1}{5})}$

Step 1.2.5.2.1.2.2

Combine $- 4$ and $\frac{1}{5}$ .

$e = - \frac{34}{5} - \frac{\frac{36}{25}}{\frac{- 4}{5}}$

Step 1.2.5.2.1.3

Move the negative in front of the fraction.

$e = - \frac{34}{5} - \frac{\frac{36}{25}}{- \frac{4}{5}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{34}{5} - (\frac{36}{25} (- \frac{5}{4}))$

Step 1.2.5.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.5.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = - \frac{34}{5} - (\frac{36}{25} \cdot \frac{- 5}{4})$

Step 1.2.5.2.1.5.2

Factor $4$ out of $36$ .

$e = - \frac{34}{5} - (\frac{4 (9)}{25} \cdot \frac{- 5}{4})$

Step 1.2.5.2.1.5.3

Cancel the common factor.

$e = - \frac{34}{5} - (\frac{4 \cdot 9}{25} \cdot \frac{- 5}{4})$

Step 1.2.5.2.1.5.4

Rewrite the expression.

$e = - \frac{34}{5} - (\frac{9}{25} \cdot - 5)$

Step 1.2.5.2.1.6

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.6.1

Factor $5$ out of $25$ .

$e = - \frac{34}{5} - (\frac{9}{5 (5)} \cdot - 5)$

Step 1.2.5.2.1.6.2

Factor $5$ out of $- 5$ .

$e = - \frac{34}{5} - (\frac{9}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.5.2.1.6.3

Cancel the common factor.

$e = - \frac{34}{5} - (\frac{9}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.5.2.1.6.4

Rewrite the expression.

$e = - \frac{34}{5} - (\frac{9}{5} \cdot - 1)$

Step 1.2.5.2.1.7

Combine $\frac{9}{5}$ and $- 1$ .

$e = - \frac{34}{5} - \frac{9 \cdot - 1}{5}$

Step 1.2.5.2.1.8

Multiply $9$ by $- 1$ .

$e = - \frac{34}{5} - \frac{- 9}{5}$

Step 1.2.5.2.1.9

Move the negative in front of the fraction.

$e = - \frac{34}{5} - - \frac{9}{5}$

Step 1.2.5.2.1.10

Multiply $- - \frac{9}{5}$ .

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Step 1.2.5.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{34}{5} + 1 (\frac{9}{5})$

Step 1.2.5.2.1.10.2

Multiply $\frac{9}{5}$ by $1$ .

$e = - \frac{34}{5} + \frac{9}{5}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 34 + 9}{5}$

Step 1.2.5.2.3

Add $- 34$ and $9$ .

$e = \frac{- 25}{5}$

Step 1.2.5.2.4

Divide $- 25$ by $5$ .

$e = - 5$

Step 1.2.6

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

$- \frac{1}{5} {(x - 3)}^{2} - 5$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{5}$

$h = 3$

$k = - 5$

Step 3

Find the vertex $(h, k)$ .

$(3, - 5)$

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.

$\frac{1}{4 \cdot (- \frac{1}{5})}$

Step 4.3

Simplify.

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Step 4.3.1

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

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Step 4.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{5})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{5})}$

Step 4.3.2

Combine $4$ and $\frac{1}{5}$ .

$- \frac{1}{\frac{4}{5}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{4}))$

Step 4.3.4

Multiply $\frac{5}{4}$ by $1$ .

$- \frac{5}{4}$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 5.2

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

$(3, - \frac{25}{4})$

Step 6