Algebra Examples

$y = \frac{2}{3} x^{2} - \frac{4}{5} x + \frac{7}{9}$

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 $\frac{2}{3}$ and $x^{2}$ .

$y = \frac{2 x^{2}}{3} - \frac{4}{5} x + \frac{7}{9}$

Step 1.1.2

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

$y = \frac{2 x^{2}}{3} - \frac{x \cdot 4}{5} + \frac{7}{9}$

Step 1.1.3

Move $4$ to the left of $x$ .

$y = \frac{2 x^{2}}{3} - \frac{4 x}{5} + \frac{7}{9}$

Step 1.2

Complete the square for $\frac{2 x^{2}}{3} - \frac{4 x}{5} + \frac{7}{9}$ .

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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 $c$ .

$a = \frac{2}{3}$

$b = - \frac{4}{5}$

$c = \frac{7}{9}$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{4}{5} \cdot \frac{1}{2 (\frac{2}{3})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

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

$d = \frac{- 4}{5} \cdot \frac{1}{2 (\frac{2}{3})}$

Step 1.2.3.2.2.2

Factor $2$ out of $- 4$ .

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

Step 1.2.3.2.2.3

Cancel the common factor.

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

Step 1.2.3.2.2.4

Rewrite the expression.

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

Step 1.2.3.2.3

Multiply $\frac{- 2}{5}$ by $\frac{1}{\frac{2}{3}}$ .

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

Step 1.2.3.2.4

Combine $5$ and $\frac{2}{3}$ .

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

Step 1.2.3.2.5

Multiply $5$ by $2$ .

$d = \frac{- 2}{\frac{10}{3}}$

Step 1.2.3.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = - 2 (\frac{3}{10})$

Step 1.2.3.2.7

Cancel the common factor of $2$ .

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Step 1.2.3.2.7.1

Factor $2$ out of $- 2$ .

$d = 2 (- 1) \frac{3}{10}$

Step 1.2.3.2.7.2

Factor $2$ out of $10$ .

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

Step 1.2.3.2.7.3

Cancel the common factor.

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

Step 1.2.3.2.7.4

Rewrite the expression.

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

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

$e = \frac{7}{9} - \frac{{(- \frac{4}{5})}^{2}}{4 (\frac{2}{3})}$

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

$e = \frac{7}{9} - \frac{{(- 1)}^{2} {(\frac{4}{5})}^{2}}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.1.2

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

$e = \frac{7}{9} - \frac{1 {(\frac{4}{5})}^{2}}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{4}{5}$ .

$e = \frac{7}{9} - \frac{1 \frac{4^{2}}{5^{2}}}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.1.4

Raise $4$ to the power of $2$ .

$e = \frac{7}{9} - \frac{1 \frac{16}{5^{2}}}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.1.5

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

$e = \frac{7}{9} - \frac{1 (\frac{16}{25})}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{16}{25}$ by $1$ .

$e = \frac{7}{9} - \frac{\frac{16}{25}}{4 (\frac{2}{3})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{2}{3}$ .

$e = \frac{7}{9} - \frac{\frac{16}{25}}{\frac{4 \cdot 2}{3}}$

Step 1.2.4.2.1.3

Multiply $4$ by $2$ .

$e = \frac{7}{9} - \frac{\frac{16}{25}}{\frac{8}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{7}{9} - (\frac{16}{25} \cdot \frac{3}{8})$

Step 1.2.4.2.1.5

Cancel the common factor of $8$ .

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Step 1.2.4.2.1.5.1

Factor $8$ out of $16$ .

$e = \frac{7}{9} - (\frac{8 (2)}{25} \cdot \frac{3}{8})$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = \frac{7}{9} - (\frac{8 \cdot 2}{25} \cdot \frac{3}{8})$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = \frac{7}{9} - (\frac{2}{25} \cdot 3)$

Step 1.2.4.2.1.6

Combine $\frac{2}{25}$ and $3$ .

$e = \frac{7}{9} - \frac{2 \cdot 3}{25}$

Step 1.2.4.2.1.7

Multiply $2$ by $3$ .

$e = \frac{7}{9} - \frac{6}{25}$

Step 1.2.4.2.2

To write $\frac{7}{9}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$e = \frac{7}{9} \cdot \frac{25}{25} - \frac{6}{25}$

Step 1.2.4.2.3

To write $- \frac{6}{25}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$e = \frac{7}{9} \cdot \frac{25}{25} - \frac{6}{25} \cdot \frac{9}{9}$

Step 1.2.4.2.4

Write each expression with a common denominator of $225$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.2.4.1

Multiply $\frac{7}{9}$ by $\frac{25}{25}$ .

$e = \frac{7 \cdot 25}{9 \cdot 25} - \frac{6}{25} \cdot \frac{9}{9}$

Step 1.2.4.2.4.2

Multiply $9$ by $25$ .

$e = \frac{7 \cdot 25}{225} - \frac{6}{25} \cdot \frac{9}{9}$

Step 1.2.4.2.4.3

Multiply $\frac{6}{25}$ by $\frac{9}{9}$ .

$e = \frac{7 \cdot 25}{225} - \frac{6 \cdot 9}{25 \cdot 9}$

Step 1.2.4.2.4.4

Multiply $25$ by $9$ .

$e = \frac{7 \cdot 25}{225} - \frac{6 \cdot 9}{225}$

Step 1.2.4.2.5

Combine the numerators over the common denominator.

$e = \frac{7 \cdot 25 - 6 \cdot 9}{225}$

Step 1.2.4.2.6

Simplify the numerator.

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Step 1.2.4.2.6.1

Multiply $7$ by $25$ .

$e = \frac{175 - 6 \cdot 9}{225}$

Step 1.2.4.2.6.2

Multiply $- 6$ by $9$ .

$e = \frac{175 - 54}{225}$

Step 1.2.4.2.6.3

Subtract $54$ from $175$ .

$e = \frac{121}{225}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{2}{3} {(x - \frac{3}{5})}^{2} + \frac{121}{225}$ .

$\frac{2}{3} {(x - \frac{3}{5})}^{2} + \frac{121}{225}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{2}{3} \cdot {(x - \frac{3}{5})}^{2} + \frac{121}{225}$

Step 2

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

$a = \frac{2}{3}$

$h = \frac{3}{5}$

$k = \frac{121}{225}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{3}{5}, \frac{121}{225})$

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.

$\frac{1}{4 \cdot \frac{2}{3}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{2}{3}$ .

$\frac{1}{\frac{4 \cdot 2}{3}}$

Step 4.3.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{8}{3}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{8})$

Step 4.3.4

Multiply $\frac{3}{8}$ by $1$ .

$\frac{3}{8}$

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.

$y = \frac{293}{1800}$

Step 6