Write in Standard Form 16x^2-224x=9y^2+18y-631

$16 x^{2} - 224 x = 9 y^{2} + 18 y - 631$

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

Subtract $9 y^{2}$ from both sides of the equation.

$16 x^{2} - 224 x - 9 y^{2} = 18 y - 631$

Subtract $18 y$ from both sides of the equation.

$16 x^{2} - 224 x - 9 y^{2} - 18 y = - 631$

Move $- 224 x$ .

$16 x^{2} - 9 y^{2} - 224 x - 18 y = - 631$

Complete the square for $16 x^{2} - 224 x$ .

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

$a = 16$

$b = - 224$

$c = 0$

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

$d = \frac{- 224}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $- 224$ and $2$ .

Factor $2$ out of $- 224$ .

$d = \frac{2 \cdot - 112}{2 \cdot 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 \cdot - 112}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot - 112}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{- 112}{16}$

Cancel the common factor of $- 112$ and $16$ .

Factor $16$ out of $- 112$ .

$d = \frac{16 \cdot - 7}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 7}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot - 7}{16 \cdot 1}$

Rewrite the expression.

$d = \frac{- 7}{1}$

Divide $- 7$ by $1$ .

$d = - 7$

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

$e = 0 - \frac{{(- 224)}^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $- 224$ to the power of $2$ .

$e = 0 - \frac{50176}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{50176}{64}$

Divide $50176$ by $64$ .

$e = 0 - 1 \cdot 784$

Multiply $- 1$ by $784$ .

$e = 0 - 784$

Subtract $784$ from $0$ .

$e = - 784$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 7)}^{2} - 784$ .

$16 {(x - 7)}^{2} - 784$

Substitute $16 {(x - 7)}^{2} - 784$ for $16 x^{2} - 224 x$ in the equation $16 x^{2} - 9 y^{2} - 224 x - 18 y = - 631$ .

$16 {(x - 7)}^{2} - 784 - 9 y^{2} - 18 y = - 631$

Move $- 784$ to the right side of the equation by adding $784$ to both sides.

$16 {(x - 7)}^{2} - 9 y^{2} - 18 y = - 631 + 784$

Complete the square for $- 9 y^{2} - 18 y$ .

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

$a = - 9$

$b = - 18$

$c = 0$

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

$d = \frac{- 18}{2 \cdot - 9}$

Simplify the right side.

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 9$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot - 9}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot - 9}$

Multiply $4$ by $- 9$ .

$e = 0 - \frac{324}{- 36}$

Divide $324$ by $- 36$ .

$e = 0 - - 9$

Multiply $- 1$ by $- 9$ .

$e = 0 + 9$

Add $0$ and $9$ .

$e = 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 9 {(y + 1)}^{2} + 9$ .

$- 9 {(y + 1)}^{2} + 9$

Substitute $- 9 {(y + 1)}^{2} + 9$ for $- 9 y^{2} - 18 y$ in the equation $16 x^{2} - 9 y^{2} - 224 x - 18 y = - 631$ .

$16 {(x - 7)}^{2} - 9 {(y + 1)}^{2} + 9 = - 631 + 784$

Move $9$ to the right side of the equation by adding $9$ to both sides.

$16 {(x - 7)}^{2} - 9 {(y + 1)}^{2} = - 631 + 784 - 9$

Simplify $- 631 + 784 - 9$ .

Add $- 631$ and $784$ .

$16 {(x - 7)}^{2} - 9 {(y + 1)}^{2} = 153 - 9$

Subtract $9$ from $153$ .

$16 {(x - 7)}^{2} - 9 {(y + 1)}^{2} = 144$

Divide each term by $144$ to make the right side equal to one.

$\frac{16 {(x - 7)}^{2}}{144} - \frac{9 {(y + 1)}^{2}}{144} = \frac{144}{144}$

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 $1$ .

$\frac{{(x - 7)}^{2}}{9} - \frac{{(y + 1)}^{2}}{16} = 1$

Write in Standard Form 16x^2-224x=9y^2+18y-631

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