Algebra Examples

$y + 7.5 = 2 {(x + 2.5)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Subtract $7.5$ from both sides of the equation.

$y = 2 {(x + 2.5)}^{2} - 7.5$

Step 1.2

Complete the square for $2 {(x + 2.5)}^{2} - 7.5$ .

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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 ${(x + 2.5)}^{2}$ as $(x + 2.5) (x + 2.5)$ .

$2 ((x + 2.5) (x + 2.5)) - 7.5$

Step 1.2.1.1.2

Expand $(x + 2.5) (x + 2.5)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$2 (x (x + 2.5) + 2.5 (x + 2.5)) - 7.5$

Step 1.2.1.1.2.2

Apply the distributive property.

$2 (x \cdot x + x \cdot 2.5 + 2.5 (x + 2.5)) - 7.5$

Step 1.2.1.1.2.3

Apply the distributive property.

$2 (x \cdot x + x \cdot 2.5 + 2.5 x + 2.5 \cdot 2.5) - 7.5$

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 $x$ .

$2 (x^{2} + x \cdot 2.5 + 2.5 x + 2.5 \cdot 2.5) - 7.5$

Step 1.2.1.1.3.1.2

Move $2.5$ to the left of $x$ .

$2 (x^{2} + 2.5 \cdot x + 2.5 x + 2.5 \cdot 2.5) - 7.5$

Step 1.2.1.1.3.1.3

Multiply $2.5$ by $2.5$ .

$2 (x^{2} + 2.5 x + 2.5 x + 6.25) - 7.5$

Step 1.2.1.1.3.2

Add $2.5 x$ and $2.5 x$ .

$2 (x^{2} + 5 x + 6.25) - 7.5$

Step 1.2.1.1.4

Apply the distributive property.

$2 x^{2} + 2 (5 x) + 2 \cdot 6.25 - 7.5$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $5$ by $2$ .

$2 x^{2} + 10 x + 2 \cdot 6.25 - 7.5$

Step 1.2.1.1.5.2

Multiply $2$ by $6.25$ .

$2 x^{2} + 10 x + 12.5 - 7.5$

Step 1.2.1.2

Subtract $7.5$ from $12.5$ .

$2 x^{2} + 10 x + 5$

Step 1.2.2

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

$a = 2$

$b = 10$

$c = 5$

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

$d = \frac{10}{2 \cdot 2}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $10$ and $2$ .

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Step 1.2.4.2.1.1

Factor $2$ out of $10$ .

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

Step 1.2.4.2.1.2

Cancel the common factors.

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Step 1.2.4.2.1.2.1

Factor $2$ out of $2 \cdot 2$ .

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

Step 1.2.4.2.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.2

Cancel the common factor of $5$ and $2$ .

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Step 1.2.4.2.2.1

Rewrite $5$ as $1 (5)$ .

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

Step 1.2.4.2.2.2

Cancel the common factors.

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Step 1.2.4.2.2.2.1

Rewrite $2$ as $1 (2)$ .

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

Step 1.2.4.2.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.2.2.3

Rewrite the expression.

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

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

$e = 5 - \frac{10^{2}}{4 \cdot 2}$

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

Raise $10$ to the power of $2$ .

$e = 5 - \frac{100}{4 \cdot 2}$

Step 1.2.5.2.1.2

Multiply $4$ by $2$ .

$e = 5 - \frac{100}{8}$

Step 1.2.5.2.1.3

Cancel the common factor of $100$ and $8$ .

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Step 1.2.5.2.1.3.1

Factor $4$ out of $100$ .

$e = 5 - \frac{4 (25)}{8}$

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 $4$ out of $8$ .

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

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

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

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

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

Step 1.2.5.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.5.2.3

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

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

Step 1.2.5.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $5$ by $2$ .

$e = \frac{10 - 25}{2}$

Step 1.2.5.2.5.2

Subtract $25$ from $10$ .

$e = \frac{- 15}{2}$

Step 1.2.5.2.6

Cancel the common factor of $- 15$ and $2$ .

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Step 1.2.5.2.6.1

Rewrite $- 15$ as $1 (- 15)$ .

$e = \frac{1 (- 15)}{2}$

Step 1.2.5.2.6.2

Cancel the common factors.

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Step 1.2.5.2.6.2.1

Rewrite $2$ as $1 (2)$ .

$e = \frac{1 \cdot - 15}{1 \cdot 2}$

Step 1.2.5.2.6.2.2

Cancel the common factor.

$e = \frac{1 \cdot - 15}{1 \cdot 2}$

Step 1.2.5.2.6.2.3

Rewrite the expression.

$e = \frac{- 15}{2}$

Step 1.2.5.2.7

Move the negative in front of the fraction.

$e = - \frac{15}{2}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{5}{2})}^{2} - \frac{15}{2}$ .

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

Step 1.3

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{5}{2})}^{2} - \frac{15}{2}$

Step 2

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

$a = 2$

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

$k = - \frac{15}{2}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{2}, - \frac{15}{2})$

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.

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

Step 4.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$y = - \frac{61}{8}$

Step 6