Precalculus Examples

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

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 $4 e (y - e) = {(x - e)}^{2}$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

Step 1.1.2.2.2

Cancel the common factor of $e$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

Divide $y - e$ by $1$ .

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

Step 1.1.3

Add $e$ to both sides of the equation.

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

Step 1.1.4

Reorder terms.

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

Step 1.2

Complete the square for $\frac{1}{4 e} \cdot {(x - e)}^{2} + e$ .

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

$\frac{1}{4 e} \cdot ((x - e) (x - e)) + e$

Step 1.2.1.1.2

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

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{4 e} \cdot (x (x - e) - e (x - e)) + e$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{4 e} \cdot (x \cdot x + x (- e) - e (x - e)) + e$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{4 e} \cdot (x \cdot x + x (- e) - e x - e (- e)) + e$

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 $x$ .

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

Step 1.2.1.1.3.1.2

Multiply $- e (- e)$ .

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Step 1.2.1.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.1.1.3.1.2.2

Multiply $e$ by $1$ .

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

Step 1.2.1.1.3.1.2.3

Raise $e$ to the power of $1$ .

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

Step 1.2.1.1.3.1.2.4

Raise $e$ to the power of $1$ .

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

Step 1.2.1.1.3.1.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.1.1.3.1.2.6

Add $1$ and $1$ .

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

Step 1.2.1.1.3.2

Move $e$ .

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

Step 1.2.1.1.3.3

Subtract $x e$ from $- 1 \cdot x e$ .

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

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{4 e} x^{2} + \frac{1}{4 e} (- 2 x e) + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $\frac{1}{4 e}$ and $x^{2}$ .

$\frac{x^{2}}{4 e} + \frac{1}{4 e} (- 2 x e) + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5.2

Cancel the common factor of $2 e$ .

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Step 1.2.1.1.5.2.1

Factor $2 e$ out of $4 e$ .

$\frac{x^{2}}{4 e} + \frac{1}{2 e (2)} (- 2 x e) + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5.2.2

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

$\frac{x^{2}}{4 e} + \frac{1}{2 e (2)} (2 e (- x)) + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5.2.3

Cancel the common factor.

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

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{x^{2}}{4 e} + \frac{1}{2} (- x) + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5.3

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

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{1}{4 e} e^{2} + e$

Step 1.2.1.1.5.4

Cancel the common factor of $e$ .

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Step 1.2.1.1.5.4.1

Factor $e$ out of $4 e$ .

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{1}{e \cdot 4} e^{2} + e$

Step 1.2.1.1.5.4.2

Factor $e$ out of $e^{2}$ .

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

Step 1.2.1.1.5.4.3

Cancel the common factor.

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

Step 1.2.1.1.5.4.4

Rewrite the expression.

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{1}{4} e + e$

Step 1.2.1.1.5.5

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

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{e}{4} + e$

Step 1.2.1.2

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

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{e}{4} + e \cdot \frac{4}{4}$

Step 1.2.1.3

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

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{e}{4} + \frac{e \cdot 4}{4}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{e + e \cdot 4}{4}$

Step 1.2.1.5

Add $e$ and $e \cdot 4$ .

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Step 1.2.1.5.1

Reorder $e$ and $4$ .

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{e + 4 \cdot e}{4}$

Step 1.2.1.5.2

Add $e$ and $4 \cdot e$ .

$\frac{x^{2}}{4 e} - \frac{x}{2} + \frac{5 e}{4}$

Step 1.2.2

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

$a = \frac{1}{4 e}$

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

$c = \frac{5 e}{4}$

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

$d = \frac{- \frac{1}{2}}{2 \frac{1}{4 e}}$

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.

$d = - \frac{1}{2} \cdot \frac{1}{2 \frac{1}{4 e}}$

Step 1.2.4.2.2

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

$d = - \frac{1}{2} \cdot \frac{1}{\frac{2}{4 e}}$

Step 1.2.4.2.3

Reduce the expression $\frac{2}{4 e}$ by cancelling the common factors.

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Step 1.2.4.2.3.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.3.2

Factor $2$ out of $4 e$ .

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

Step 1.2.4.2.3.3

Cancel the common factor.

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

Step 1.2.4.2.3.4

Rewrite the expression.

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

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.2.4.2.5.2

Factor $2$ out of $1 (2 e)$ .

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

Step 1.2.4.2.5.3

Cancel the common factor.

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

Step 1.2.4.2.5.4

Rewrite the expression.

$d = - (1 (e))$

Step 1.2.4.2.6

Multiply $e$ by $1$ .

$d = - e$

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

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

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

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

Step 1.2.5.2.1.1.2

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

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

Step 1.2.5.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2.5.2.1.1.4

One to any power is one.

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

Step 1.2.5.2.1.1.5

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

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

Step 1.2.5.2.1.1.6

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

$e = \frac{5 e}{4} - \frac{\frac{1}{4}}{4 \frac{1}{4 e}}$

Step 1.2.5.2.1.2

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

$e = \frac{5 e}{4} - \frac{\frac{1}{4}}{\frac{4}{4 e}}$

Step 1.2.5.2.1.3

Reduce the expression $\frac{4}{4 e}$ by cancelling the common factors.

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Step 1.2.5.2.1.3.1

Cancel the common factor.

$e = \frac{5 e}{4} - \frac{\frac{1}{4}}{\frac{4}{4 e}}$

Step 1.2.5.2.1.3.2

Rewrite the expression.

$e = \frac{5 e}{4} - \frac{\frac{1}{4}}{\frac{1}{e}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.5.2.1.5

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

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

Step 1.2.5.2.2

Combine the numerators over the common denominator.

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

Step 1.2.5.2.3

Subtract $e$ from $5 e$ .

$e = \frac{4 e}{4}$

Step 1.2.5.2.4

Cancel the common factor of $4$ .

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Step 1.2.5.2.4.1

Cancel the common factor.

$e = \frac{4 e}{4}$

Step 1.2.5.2.4.2

Divide $e$ by $1$ .

$e = e$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4 e} {(x - e)}^{2} + e$ .

$\frac{1}{4 e} {(x - e)}^{2} + e$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = \frac{1}{4 e}$

$h = e$

$k = e$

Step 3

Find the vertex $(h, k)$ .

$(e, e)$

Step 4