Algebra Examples

${(x - 2)}^{2} = 10 (y - 3)$

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

Rewrite the equation as $10 (y - 3) = {(x - 2)}^{2}$ .

$10 (y - 3) = {(x - 2)}^{2}$

Step 1.1.2

Divide each term in $10 (y - 3) = {(x - 2)}^{2}$ by $10$ and simplify.

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Step 1.1.2.1

Divide each term in $10 (y - 3) = {(x - 2)}^{2}$ by $10$ .

$\frac{10 (y - 3)}{10} = \frac{{(x - 2)}^{2}}{10}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $10$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{10 (y - 3)}{10} = \frac{{(x - 2)}^{2}}{10}$

Step 1.1.2.2.1.2

Divide $y - 3$ by $1$ .

$y - 3 = \frac{{(x - 2)}^{2}}{10}$

Step 1.1.3

Add $3$ to both sides of the equation.

$y = \frac{{(x - 2)}^{2}}{10} + 3$

Step 1.1.4

Reorder terms.

$y = \frac{1}{10} \cdot {(x - 2)}^{2} + 3$

Step 1.2

Complete the square for $\frac{1}{10} \cdot {(x - 2)}^{2} + 3$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$\frac{1}{10} \cdot ((x - 2) (x - 2)) + 3$

Step 1.2.1.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{10} \cdot (x (x - 2) - 2 (x - 2)) + 3$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{10} \cdot (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 3$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{10} \cdot (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 3$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{1}{10} \cdot (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + 3$

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $x$ .

$\frac{1}{10} \cdot (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + 3$

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{1}{10} \cdot (x^{2} - 2 x - 2 x + 4) + 3$

Step 1.2.1.1.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{1}{10} \cdot (x^{2} - 4 x + 4) + 3$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{10} x^{2} + \frac{1}{10} (- 4 x) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $\frac{1}{10}$ and $x^{2}$ .

$\frac{x^{2}}{10} + \frac{1}{10} (- 4 x) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.2.1

Factor $2$ out of $10$ .

$\frac{x^{2}}{10} + \frac{1}{2 (5)} (- 4 x) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.2.2

Factor $2$ out of $- 4 x$ .

$\frac{x^{2}}{10} + \frac{1}{2 (5)} (2 (- 2 x)) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{x^{2}}{10} + \frac{1}{2 \cdot 5} (2 (- 2 x)) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{x^{2}}{10} + \frac{1}{5} (- 2 x) + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.3

Combine $- 2$ and $\frac{1}{5}$ .

$\frac{x^{2}}{10} + \frac{- 2}{5} x + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.4

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

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{1}{10} \cdot 4 + 3$

Step 1.2.1.1.5.5

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.5.1

Factor $2$ out of $10$ .

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{1}{2 (5)} \cdot 4 + 3$

Step 1.2.1.1.5.5.2

Factor $2$ out of $4$ .

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{1}{2 \cdot 5} \cdot (2 \cdot 2) + 3$

Step 1.2.1.1.5.5.3

Cancel the common factor.

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{1}{2 \cdot 5} \cdot (2 \cdot 2) + 3$

Step 1.2.1.1.5.5.4

Rewrite the expression.

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{1}{5} \cdot 2 + 3$

Step 1.2.1.1.5.6

Combine $\frac{1}{5}$ and $2$ .

$\frac{x^{2}}{10} + \frac{- 2 x}{5} + \frac{2}{5} + 3$

Step 1.2.1.1.6

Move the negative in front of the fraction.

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{2}{5} + 3$

Step 1.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{2}{5} + 3 \cdot \frac{5}{5}$

Step 1.2.1.3

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

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{2}{5} + \frac{3 \cdot 5}{5}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{2 + 3 \cdot 5}{5}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $3$ by $5$ .

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{2 + 15}{5}$

Step 1.2.1.5.2

Add $2$ and $15$ .

$\frac{x^{2}}{10} - \frac{2 x}{5} + \frac{17}{5}$

Step 1.2.2

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

$a = \frac{1}{10}$

$b = - \frac{2}{5}$

$c = \frac{17}{5}$

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

$d = \frac{- \frac{2}{5}}{2 (\frac{1}{10})}$

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.

$d = - \frac{2}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$d = \frac{- 2}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Step 1.2.4.2.2.2

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Step 1.2.4.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot - 1}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Step 1.2.4.2.2.4

Rewrite the expression.

$d = \frac{- 1}{5} \cdot \frac{1}{\frac{1}{10}}$

Step 1.2.4.2.3

Multiply $\frac{- 1}{5}$ by $\frac{1}{\frac{1}{10}}$ .

$d = \frac{- 1}{5 (\frac{1}{10})}$

Step 1.2.4.2.4

Combine $5$ and $\frac{1}{10}$ .

$d = \frac{- 1}{\frac{5}{10}}$

Step 1.2.4.2.5

Cancel the common factor of $5$ and $10$ .

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Step 1.2.4.2.5.1

Factor $5$ out of $5$ .

$d = \frac{- 1}{\frac{5 (1)}{10}}$

Step 1.2.4.2.5.2

Cancel the common factors.

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Step 1.2.4.2.5.2.1

Factor $5$ out of $10$ .

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

Step 1.2.4.2.5.2.2

Cancel the common factor.

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

Step 1.2.4.2.5.2.3

Rewrite the expression.

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

Step 1.2.4.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Step 1.2.4.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

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

$e = \frac{17}{5} - \frac{{(- \frac{2}{5})}^{2}}{4 (\frac{1}{10})}$

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

$e = \frac{17}{5} - \frac{{(- 1)}^{2} {(\frac{2}{5})}^{2}}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.1.2

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

$e = \frac{17}{5} - \frac{1 {(\frac{2}{5})}^{2}}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.1.3

Apply the product rule to $\frac{2}{5}$ .

$e = \frac{17}{5} - \frac{1 \frac{2^{2}}{5^{2}}}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.1.4

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

$e = \frac{17}{5} - \frac{1 \frac{4}{5^{2}}}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.1.5

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

$e = \frac{17}{5} - \frac{1 (\frac{4}{25})}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.1.6

Multiply $\frac{4}{25}$ by $1$ .

$e = \frac{17}{5} - \frac{\frac{4}{25}}{4 (\frac{1}{10})}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{1}{10}$ .

$e = \frac{17}{5} - \frac{\frac{4}{25}}{\frac{4}{10}}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ and $10$ .

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Step 1.2.5.2.1.3.1

Factor $2$ out of $4$ .

$e = \frac{17}{5} - \frac{\frac{4}{25}}{\frac{2 (2)}{10}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $2$ out of $10$ .

$e = \frac{17}{5} - \frac{\frac{4}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = \frac{17}{5} - \frac{\frac{4}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = \frac{17}{5} - \frac{\frac{4}{25}}{\frac{2}{5}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{17}{5} - (\frac{4}{25} \cdot \frac{5}{2})$

Step 1.2.5.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.5.1

Factor $2$ out of $4$ .

$e = \frac{17}{5} - (\frac{2 (2)}{25} \cdot \frac{5}{2})$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{17}{5} - (\frac{2 \cdot 2}{25} \cdot \frac{5}{2})$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{17}{5} - (\frac{2}{25} \cdot 5)$

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 $25$ .

$e = \frac{17}{5} - (\frac{2}{5 (5)} \cdot 5)$

Step 1.2.5.2.1.6.2

Cancel the common factor.

$e = \frac{17}{5} - (\frac{2}{5 \cdot 5} \cdot 5)$

Step 1.2.5.2.1.6.3

Rewrite the expression.

$e = \frac{17}{5} - \frac{2}{5}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{17 - 2}{5}$

Step 1.2.5.2.3

Subtract $2$ from $17$ .

$e = \frac{15}{5}$

Step 1.2.5.2.4

Divide $15$ by $5$ .

$e = 3$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{10} {(x - 2)}^{2} + 3$ .

$\frac{1}{10} {(x - 2)}^{2} + 3$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{10} \cdot {(x - 2)}^{2} + 3$

Step 2

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

$a = \frac{1}{10}$

$h = 2$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(2, 3)$

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.

$\frac{1}{4 \cdot \frac{1}{10}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{1}{10}$ .

$\frac{1}{\frac{4}{10}}$

Step 4.3.2

Cancel the common factor of $4$ and $10$ .

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Step 4.3.2.1

Factor $2$ out of $4$ .

$\frac{1}{\frac{2 (2)}{10}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $2$ out of $10$ .

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

Step 4.3.2.2.2

Cancel the common factor.

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

Step 4.3.2.2.3

Rewrite the expression.

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

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.4

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

$\frac{5}{2}$

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.

$(2, \frac{11}{2})$

Step 6