Algebra Examples

$y = - \frac{5}{8} x^{2} - 3 x + 4$

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 $\frac{5}{8}$ .

$y = - \frac{x^{2} \cdot 5}{8} - 3 x + 4$

Step 1.1.2

Move $5$ to the left of $x^{2}$ .

$y = - \frac{5 x^{2}}{8} - 3 x + 4$

Step 1.2

Complete the square for $- \frac{5 x^{2}}{8} - 3 x + 4$ .

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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 $c$ .

$a = - \frac{5}{8}$

$b = - 3$

$c = 4$

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

$d = \frac{- 3}{2 (- \frac{5}{8})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify the denominator.

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Step 1.2.3.2.1.1

Multiply $- 1$ by $2$ .

$d = \frac{- 3}{- 2 (\frac{5}{8})}$

Step 1.2.3.2.1.2

Combine $- 2$ and $\frac{5}{8}$ .

$d = \frac{- 3}{\frac{- 2 \cdot 5}{8}}$

Step 1.2.3.2.2

Multiply $- 2$ by $5$ .

$d = \frac{- 3}{\frac{- 10}{8}}$

Step 1.2.3.2.3

Reduce the expression by cancelling the common factors.

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Step 1.2.3.2.3.1

Cancel the common factor of $- 10$ and $8$ .

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Step 1.2.3.2.3.1.1

Factor $2$ out of $- 10$ .

$d = \frac{- 3}{\frac{2 (- 5)}{8}}$

Step 1.2.3.2.3.1.2

Cancel the common factors.

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Step 1.2.3.2.3.1.2.1

Factor $2$ out of $8$ .

$d = \frac{- 3}{\frac{2 \cdot - 5}{2 \cdot 4}}$

Step 1.2.3.2.3.1.2.2

Cancel the common factor.

$d = \frac{- 3}{\frac{2 \cdot - 5}{2 \cdot 4}}$

Step 1.2.3.2.3.1.2.3

Rewrite the expression.

$d = \frac{- 3}{\frac{- 5}{4}}$

Step 1.2.3.2.3.2

Move the negative in front of the fraction.

$d = \frac{- 3}{- \frac{5}{4}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - 3 (- \frac{4}{5})$

Step 1.2.3.2.5

Multiply $- 3 (- \frac{4}{5})$ .

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Step 1.2.3.2.5.1

Multiply $- 1$ by $- 3$ .

$d = 3 (\frac{4}{5})$

Step 1.2.3.2.5.2

Combine $3$ and $\frac{4}{5}$ .

$d = \frac{3 \cdot 4}{5}$

Step 1.2.3.2.5.3

Multiply $3$ by $4$ .

$d = \frac{12}{5}$

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

$e = 4 - \frac{{(- 3)}^{2}}{4 (- \frac{5}{8})}$

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

Raise $- 3$ to the power of $2$ .

$e = 4 - \frac{9}{4 (- \frac{5}{8})}$

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 $4$ .

$e = 4 - \frac{9}{- 4 (\frac{5}{8})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{5}{8}$ .

$e = 4 - \frac{9}{\frac{- 4 \cdot 5}{8}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $5$ .

$e = 4 - \frac{9}{\frac{- 20}{8}}$

Step 1.2.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.4.1

Cancel the common factor of $- 20$ and $8$ .

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Step 1.2.4.2.1.4.1.1

Factor $4$ out of $- 20$ .

$e = 4 - \frac{9}{\frac{4 (- 5)}{8}}$

Step 1.2.4.2.1.4.1.2

Cancel the common factors.

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Step 1.2.4.2.1.4.1.2.1

Factor $4$ out of $8$ .

$e = 4 - \frac{9}{\frac{4 \cdot - 5}{4 \cdot 2}}$

Step 1.2.4.2.1.4.1.2.2

Cancel the common factor.

$e = 4 - \frac{9}{\frac{4 \cdot - 5}{4 \cdot 2}}$

Step 1.2.4.2.1.4.1.2.3

Rewrite the expression.

$e = 4 - \frac{9}{\frac{- 5}{2}}$

Step 1.2.4.2.1.4.2

Move the negative in front of the fraction.

$e = 4 - \frac{9}{- \frac{5}{2}}$

Step 1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = 4 - (9 (- \frac{2}{5}))$

Step 1.2.4.2.1.6

Multiply $9 (- \frac{2}{5})$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $9$ .

$e = 4 - (- 9 (\frac{2}{5}))$

Step 1.2.4.2.1.6.2

Combine $- 9$ and $\frac{2}{5}$ .

$e = 4 - \frac{- 9 \cdot 2}{5}$

Step 1.2.4.2.1.6.3

Multiply $- 9$ by $2$ .

$e = 4 - \frac{- 18}{5}$

Step 1.2.4.2.1.7

Move the negative in front of the fraction.

$e = 4 - - \frac{18}{5}$

Step 1.2.4.2.1.8

Multiply $- - \frac{18}{5}$ .

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Step 1.2.4.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = 4 + 1 (\frac{18}{5})$

Step 1.2.4.2.1.8.2

Multiply $\frac{18}{5}$ by $1$ .

$e = 4 + \frac{18}{5}$

Step 1.2.4.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = 4 \cdot \frac{5}{5} + \frac{18}{5}$

Step 1.2.4.2.3

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

$e = \frac{4 \cdot 5}{5} + \frac{18}{5}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{4 \cdot 5 + 18}{5}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $4$ by $5$ .

$e = \frac{20 + 18}{5}$

Step 1.2.4.2.5.2

Add $20$ and $18$ .

$e = \frac{38}{5}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{5}{8} {(x + \frac{12}{5})}^{2} + \frac{38}{5}$ .

$- \frac{5}{8} {(x + \frac{12}{5})}^{2} + \frac{38}{5}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{5}{8} \cdot {(x + \frac{12}{5})}^{2} + \frac{38}{5}$

Step 2

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

$a = - \frac{5}{8}$

$h = - \frac{12}{5}$

$k = \frac{38}{5}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{12}{5}, \frac{38}{5})$

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.

$\frac{1}{4 \cdot (- \frac{5}{8})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{5}{8})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{5}{8})}$

Step 4.3.2

Combine $4$ and $\frac{5}{8}$ .

$- \frac{1}{\frac{4 \cdot 5}{8}}$

Step 4.3.3

Multiply $4$ by $5$ .

$- \frac{1}{\frac{20}{8}}$

Step 4.3.4

Cancel the common factor of $20$ and $8$ .

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Step 4.3.4.1

Factor $4$ out of $20$ .

$- \frac{1}{\frac{4 (5)}{8}}$

Step 4.3.4.2

Cancel the common factors.

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Step 4.3.4.2.1

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 5}{4 \cdot 2}}$

Step 4.3.4.2.2

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 5}{4 \cdot 2}}$

Step 4.3.4.2.3

Rewrite the expression.

$- \frac{1}{\frac{5}{2}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.6

Multiply $\frac{2}{5}$ by $1$ .

$- \frac{2}{5}$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 5.2

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

$y = 8$

Step 6