Algebra Examples

$y = - \frac{1}{2} x^{2} - \frac{1}{5} x$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} - \frac{1}{5} x$

Step 1.1.2

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

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

Step 1.2

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

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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 $c$ .

$a = - \frac{1}{2}$

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

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{2}$ .

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

Step 1.2.3.2.4

Divide $2$ by $2$ .

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

Step 1.2.3.2.5

Cancel the common factor of $1$ .

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Step 1.2.3.2.5.1

Cancel the common factor.

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

Step 1.2.3.2.5.2

Rewrite the expression.

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

Step 1.2.3.2.6

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

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

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- \frac{1}{5})}^{2}}{4 (- \frac{1}{2})}$

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

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $- \frac{1}{5}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{1}{5})}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 {(\frac{1}{5})}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{5}$ .

$e = 0 - \frac{1 \frac{1^{2}}{5^{2}}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.4

One to any power is one.

$e = 0 - \frac{1 \frac{1}{5^{2}}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.5

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

$e = 0 - \frac{1 (\frac{1}{25})}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{1}{25}$ by $1$ .

$e = 0 - \frac{\frac{1}{25}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{\frac{1}{25}}{- 4 (\frac{1}{2})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

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

Step 1.2.4.2.1.3

Divide $- 4$ by $2$ .

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

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{1}{25} \cdot \frac{1}{- 2})$

Step 1.2.4.2.1.5

Move the negative in front of the fraction.

$e = 0 - (\frac{1}{25} (- \frac{1}{2}))$

Step 1.2.4.2.1.6

Multiply $\frac{1}{25} (- \frac{1}{2})$ .

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Step 1.2.4.2.1.6.1

Multiply $\frac{1}{25}$ by $\frac{1}{2}$ .

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

Step 1.2.4.2.1.6.2

Multiply $25$ by $2$ .

$e = 0 - - \frac{1}{50}$

Step 1.2.4.2.1.7

Multiply $- - \frac{1}{50}$ .

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Step 1.2.4.2.1.7.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{1}{50})$

Step 1.2.4.2.1.7.2

Multiply $\frac{1}{50}$ by $1$ .

$e = 0 + \frac{1}{50}$

Step 1.2.4.2.2

Add $0$ and $\frac{1}{50}$ .

$e = \frac{1}{50}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + \frac{1}{5})}^{2} + \frac{1}{50}$ .

$- \frac{1}{2} {(x + \frac{1}{5})}^{2} + \frac{1}{50}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + \frac{1}{5})}^{2} + \frac{1}{50}$

Step 2

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

$a = - \frac{1}{2}$

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

$k = \frac{1}{50}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{1}{5}, \frac{1}{50})$

Step 4