Algebra Examples

$x^{2} = 12 {(y + \frac{5}{2})}^{2}$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $12 {(y + \frac{5}{2})}^{2}$ from both sides of the equation.

$x^{2} - 12 {(y + \frac{5}{2})}^{2} = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(y + \frac{5}{2})}^{2}$ as $(y + \frac{5}{2}) (y + \frac{5}{2})$ .

$x^{2} - 12 ((y + \frac{5}{2}) (y + \frac{5}{2})) = 0$

Step 1.2.2

Expand $(y + \frac{5}{2}) (y + \frac{5}{2})$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x^{2} - 12 (y (y + \frac{5}{2}) + \frac{5}{2} (y + \frac{5}{2})) = 0$

Step 1.2.2.2

Apply the distributive property.

$x^{2} - 12 (y \cdot y + y \frac{5}{2} + \frac{5}{2} (y + \frac{5}{2})) = 0$

Step 1.2.2.3

Apply the distributive property.

$x^{2} - 12 (y \cdot y + y \frac{5}{2} + \frac{5}{2} y + \frac{5}{2} \cdot \frac{5}{2}) = 0$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $y$ by $y$ .

$x^{2} - 12 (y^{2} + y \frac{5}{2} + \frac{5}{2} y + \frac{5}{2} \cdot \frac{5}{2}) = 0$

Step 1.2.3.1.2

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

$x^{2} - 12 (y^{2} + \frac{y \cdot 5}{2} + \frac{5}{2} y + \frac{5}{2} \cdot \frac{5}{2}) = 0$

Step 1.2.3.1.3

Move $5$ to the left of $y$ .

$x^{2} - 12 (y^{2} + \frac{5 \cdot y}{2} + \frac{5}{2} y + \frac{5}{2} \cdot \frac{5}{2}) = 0$

Step 1.2.3.1.4

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

$x^{2} - 12 (y^{2} + \frac{5 y}{2} + \frac{5 y}{2} + \frac{5}{2} \cdot \frac{5}{2}) = 0$

Step 1.2.3.1.5

Multiply $\frac{5}{2} \cdot \frac{5}{2}$ .

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Step 1.2.3.1.5.1

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

$x^{2} - 12 (y^{2} + \frac{5 y}{2} + \frac{5 y}{2} + \frac{5 \cdot 5}{2 \cdot 2}) = 0$

Step 1.2.3.1.5.2

Multiply $5$ by $5$ .

$x^{2} - 12 (y^{2} + \frac{5 y}{2} + \frac{5 y}{2} + \frac{25}{2 \cdot 2}) = 0$

Step 1.2.3.1.5.3

Multiply $2$ by $2$ .

$x^{2} - 12 (y^{2} + \frac{5 y}{2} + \frac{5 y}{2} + \frac{25}{4}) = 0$

Step 1.2.3.2

Add $\frac{5 y}{2}$ and $\frac{5 y}{2}$ .

$x^{2} - 12 (y^{2} + 2 \frac{5 y}{2} + \frac{25}{4}) = 0$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$x^{2} - 12 (y^{2} + 2 \frac{5 y}{2} + \frac{25}{4}) = 0$

Step 1.2.4.2

Rewrite the expression.

$x^{2} - 12 (y^{2} + 5 y + \frac{25}{4}) = 0$

Step 1.2.5

Apply the distributive property.

$x^{2} - 12 y^{2} - 12 (5 y) - 12 (\frac{25}{4}) = 0$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Multiply $5$ by $- 12$ .

$x^{2} - 12 y^{2} - 60 y - 12 (\frac{25}{4}) = 0$

Step 1.2.6.2

Cancel the common factor of $4$ .

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Step 1.2.6.2.1

Factor $4$ out of $- 12$ .

$x^{2} - 12 y^{2} - 60 y + 4 (- 3) \frac{25}{4} = 0$

Step 1.2.6.2.2

Cancel the common factor.

$x^{2} - 12 y^{2} - 60 y + 4 \cdot - 3 \frac{25}{4} = 0$

Step 1.2.6.2.3

Rewrite the expression.

$x^{2} - 12 y^{2} - 60 y - 3 \cdot 25 = 0$

Step 1.2.6.3

Multiply $- 3$ by $25$ .

$x^{2} - 12 y^{2} - 60 y - 75 = 0$

Step 2

Add $75$ to both sides of the equation.

$x^{2} - 12 y^{2} - 60 y = 75$

Step 3

Complete the square for $- 12 y^{2} - 60 y$ .

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Step 3.1

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

$a = - 12$

$b = - 60$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 60}{2 \cdot - 12}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $- 60$ and $2$ .

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Step 3.3.2.1.1

Factor $2$ out of $- 60$ .

$d = \frac{2 \cdot - 30}{2 \cdot - 12}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Factor $2$ out of $2 \cdot - 12$ .

$d = \frac{2 \cdot - 30}{2 (- 12)}$

Step 3.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 30}{2 \cdot - 12}$

Step 3.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 30}{- 12}$

Step 3.3.2.2

Cancel the common factor of $- 30$ and $- 12$ .

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Step 3.3.2.2.1

Factor $- 6$ out of $- 30$ .

$d = \frac{- 6 (5)}{- 12}$

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor $- 6$ out of $- 12$ .

$d = \frac{- 6 \cdot 5}{- 6 \cdot 2}$

Step 3.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 6 \cdot 5}{- 6 \cdot 2}$

Step 3.3.2.2.2.3

Rewrite the expression.

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

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 60)}^{2}}{4 \cdot - 12}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $- 60$ to the power of $2$ .

$e = 0 - \frac{3600}{4 \cdot - 12}$

Step 3.4.2.1.2

Multiply $4$ by $- 12$ .

$e = 0 - \frac{3600}{- 48}$

Step 3.4.2.1.3

Divide $3600$ by $- 48$ .

$e = 0 - - 75$

Step 3.4.2.1.4

Multiply $- 1$ by $- 75$ .

$e = 0 + 75$

Step 3.4.2.2

Add $0$ and $75$ .

$e = 75$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 12 {(y + \frac{5}{2})}^{2} + 75$ .

$- 12 {(y + \frac{5}{2})}^{2} + 75$

Step 4

Substitute $- 12 {(y + \frac{5}{2})}^{2} + 75$ for $- 12 y^{2} - 60 y$ in the equation $x^{2} - 12 y^{2} - 60 y = 75$ .

$x^{2} - 12 {(y + \frac{5}{2})}^{2} + 75 = 75$

Step 5

Move $75$ to the right side of the equation by adding $75$ to both sides.

$x^{2} - 12 {(y + \frac{5}{2})}^{2} = 75 - 75$

Step 6

Subtract $75$ from $75$ .

$x^{2} - 12 {(y + \frac{5}{2})}^{2} = 0$

Step 7

Divide each term by $0$ to make the right side equal to one.

$\frac{x^{2}}{0} - \frac{12 {(y + \frac{5}{2})}^{2}}{0} = \frac{0}{0}$

Step 8

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$x^{2} - {(y + \frac{5}{2})}^{2} = 1$