Precalculus Examples

$- 2 y^{2} + x - 20 y - 49 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $2 y^{2}$ to both sides of the equation.

$x - 20 y - 49 = 2 y^{2}$

Step 1.2

Add $20 y$ to both sides of the equation.

$x - 49 = 2 y^{2} + 20 y$

Step 1.3

Add $49$ to both sides of the equation.

$x = 2 y^{2} + 20 y + 49$

Step 2

Complete the square for $2 y^{2} + 20 y + 49$ .

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Step 2.1

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

$a = 2$

$b = 20$

$c = 49$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{20}{2 \cdot 2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $20$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $20$ .

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

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

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

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

Factor $2$ out of $10$ .

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $5$ by $1$ .

$d = 5$

Step 2.4

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

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Step 2.4.1

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

$e = 49 - \frac{20^{2}}{4 \cdot 2}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $20$ to the power of $2$ .

$e = 49 - \frac{400}{4 \cdot 2}$

Step 2.4.2.1.2

Multiply $4$ by $2$ .

$e = 49 - \frac{400}{8}$

Step 2.4.2.1.3

Divide $400$ by $8$ .

$e = 49 - 1 \cdot 50$

Step 2.4.2.1.4

Multiply $- 1$ by $50$ .

$e = 49 - 50$

Step 2.4.2.2

Subtract $50$ from $49$ .

$e = - 1$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(y + 5)}^{2} - 1$ .

$2 {(y + 5)}^{2} - 1$

Step 3

Set $x$ equal to the new right side.

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