Trigonometry Examples

$x^{2} + 10 x + y + 31 = 0$

Step 1

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

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$10 x + y + 31 = - x^{2}$

Step 1.2

Subtract $10 x$ from both sides of the equation.

$y + 31 = - x^{2} - 10 x$

Step 1.3

Subtract $31$ from both sides of the equation.

$y = - x^{2} - 10 x - 31$

Step 2

Complete the square for $- x^{2} - 10 x - 31$ .

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Step 2.1

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

$a = - 1$

$b = - 10$

$c = - 31$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $- 10$ .

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

Step 2.3.2.1.2

Move the negative one from the denominator of $\frac{- 5}{- 1}$ .

$d = - 1 \cdot - 5$

Step 2.3.2.2

Rewrite $- 1 \cdot - 5$ as $- - 5$ .

$d = - - 5$

Step 2.3.2.3

Multiply $- 1$ by $- 5$ .

$d = 5$

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

$e = - 31 - \frac{{(- 10)}^{2}}{4 \cdot - 1}$

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

$e = - 31 - \frac{100}{4 \cdot - 1}$

Step 2.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 31 - \frac{100}{- 4}$

Step 2.4.2.1.3

Divide $100$ by $- 4$ .

$e = - 31 - - 25$

Step 2.4.2.1.4

Multiply $- 1$ by $- 25$ .

$e = - 31 + 25$

Step 2.4.2.2

Add $- 31$ and $25$ .

$e = - 6$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 5)}^{2} - 6$ .

$- {(x + 5)}^{2} - 6$

Step 3

Set $y$ equal to the new right side.

$y = - {(x + 5)}^{2} - 6$