Algebra Examples

$8 x^{2} + 4 y = 0$

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

Subtract $8 x^{2}$ from both sides of the equation.

$4 y = - 8 x^{2}$

Step 1.1.2

Divide each term in $4 y = - 8 x^{2}$ by $4$ and simplify.

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Step 1.1.2.1

Divide each term in $4 y = - 8 x^{2}$ by $4$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

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

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Step 1.1.2.3.1.1

Factor $4$ out of $- 8 x^{2}$ .

$y = \frac{4 (- 2 x^{2})}{4}$

Step 1.1.2.3.1.2

Cancel the common factors.

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Step 1.1.2.3.1.2.1

Factor $4$ out of $4$ .

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

Step 1.1.2.3.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.2.4

Divide $- 2 x^{2}$ by $1$ .

$y = - 2 x^{2}$

Step 1.2

Complete the square for $- 2 x^{2}$ .

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Step 1.2.1

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

$a = - 2$

$b = 0$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

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

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

Factor $2$ out of $0$ .

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

Step 1.2.3.2.2.2

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

$d = - 1 \cdot 0$

Step 1.2.3.2.3

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

$d = - 0$

Step 1.2.3.2.4

Multiply $- 1$ by $0$ .

$d = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

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

Step 1.2.4.2.1.2

Multiply $4$ by $- 2$ .

$e = 0 - \frac{0}{- 8}$

Step 1.2.4.2.1.3

Divide $0$ by $- 8$ .

$e = 0 - 0$

Step 1.2.4.2.1.4

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 x^{2}$ .

$- 2 x^{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = - 2 x^{2}$

Step 2

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

$a = - 2$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

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.

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

Step 4.3

Simplify.

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Step 4.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(0, - \frac{1}{8})$

Step 6