Find the Vertex Form 162x+731=-y-9x^2

$162 x + 731 = - y - 9 x^{2}$

Step 1

Isolate $y$ to the left side of the equation.

Rewrite the equation as $- y - 9 x^{2} = 162 x + 731$ .

$- y - 9 x^{2} = 162 x + 731$

Add $9 x^{2}$ to both sides of the equation.

$- y = 162 x + 731 + 9 x^{2}$

Divide each term in $- y = 162 x + 731 + 9 x^{2}$ by $- 1$ and simplify.

Divide each term in $- y = 162 x + 731 + 9 x^{2}$ by $- 1$ .

$\frac{- y}{- 1} = \frac{162 x}{- 1} + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{162 x}{- 1} + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Divide $y$ by $1$ .

$y = \frac{162 x}{- 1} + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{162 x}{- 1}$ .

$y = - 1 \cdot (162 x) + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Rewrite $- 1 \cdot (162 x)$ as $- (162 x)$ .

$y = - (162 x) + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Multiply $162$ by $- 1$ .

$y = - 162 x + \frac{731}{- 1} + \frac{9 x^{2}}{- 1}$

Divide $731$ by $- 1$ .

$y = - 162 x - 731 + \frac{9 x^{2}}{- 1}$

Move the negative one from the denominator of $\frac{9 x^{2}}{- 1}$ .

$y = - 162 x - 731 - 1 \cdot (9 x^{2})$

Rewrite $- 1 \cdot (9 x^{2})$ as $- (9 x^{2})$ .

$y = - 162 x - 731 - (9 x^{2})$

Multiply $9$ by $- 1$ .

$y = - 162 x - 731 - 9 x^{2}$

Step 2

Complete the square for $- 162 x - 731 - 9 x^{2}$ .

Simplify the expression.

Move $- 731$ .

$- 162 x - 9 x^{2} - 731$

Reorder $- 162 x$ and $- 9 x^{2}$ .

$- 9 x^{2} - 162 x - 731$

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

$a = - 9$

$b = - 162$

$c = - 731$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- 162}{2 \cdot - 9}$

Simplify the right side.

Cancel the common factor of $- 162$ and $2$ .

Factor $2$ out of $- 162$ .

$d = \frac{2 \cdot - 81}{2 \cdot - 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 9$ .

$d = \frac{2 \cdot - 81}{2 (- 9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 81}{2 \cdot - 9}$

Rewrite the expression.

$d = \frac{- 81}{- 9}$

Cancel the common factor of $- 81$ and $- 9$ .

Factor $- 9$ out of $- 81$ .

$d = \frac{- 9 (9)}{- 9}$

Cancel the common factors.

Factor $- 9$ out of $- 9$ .

$d = \frac{- 9 \cdot 9}{- 9 \cdot 1}$

Cancel the common factor.

$d = \frac{- 9 \cdot 9}{- 9 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

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

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

$e = - 731 - \frac{{(- 162)}^{2}}{4 \cdot - 9}$

Simplify the right side.

Simplify each term.

Raise $- 162$ to the power of $2$ .

$e = - 731 - \frac{26244}{4 \cdot - 9}$

Multiply $4$ by $- 9$ .

$e = - 731 - \frac{26244}{- 36}$

Divide $26244$ by $- 36$ .

$e = - 731 - - 729$

Multiply $- 1$ by $- 729$ .

$e = - 731 + 729$

Add $- 731$ and $729$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 9 {(x + 9)}^{2} - 2$ .

$- 9 {(x + 9)}^{2} - 2$

Step 3

Set $y$ equal to the new right side.

$y = - 9 {(x + 9)}^{2} - 2$

Find the Vertex Form 162x+731=-y-9x^2

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