Write in Standard Form 9x^2+72x-64y^2+128y+80=0

$9 x^{2} + 72 x - 64 y^{2} + 128 y + 80 = 0$

Step 1

Solve for $y$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 64$ , $b = 128$ , and $c = 9 x^{2} + 72 x + 80$ into the quadratic formula and solve for $y$ .

$\frac{- 128 \pm \sqrt{128^{2} - 4 \cdot (- 64 \cdot (9 x^{2} + 72 x + 80))}}{2 \cdot - 64}$

Simplify.

Simplify the numerator.

Raise $128$ to the power of $2$ .

$y = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 64 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Multiply $- 4$ by $- 64$ .

$y = \frac{- 128 \pm \sqrt{16384 + 256 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm \sqrt{16384 + 256 (9 x^{2}) + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Simplify.

Multiply $9$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $72$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $256$ by $80$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 20480}}{2 \cdot - 64}$

Add $16384$ and $20480$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 18432 x + 36864}}{2 \cdot - 64}$

Rewrite $2304 x^{2} + 18432 x + 36864$ in a factored form.

Factor $2304$ out of $2304 x^{2} + 18432 x + 36864$ .

Factor $2304$ out of $2304 x^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 18432 x + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $18432 x$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 2304 (8 x) + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $36864$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 2304 (8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 x^{2} + 2304 (8 x)$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 (x^{2} + 8 x) + 2304 \cdot 16$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 16)}}{2 \cdot - 64}$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 4^{2})}}{2 \cdot - 64}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 2 \cdot x \cdot 4 + 4^{2})}}{2 \cdot - 64}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$y = \frac{- 128 \pm \sqrt{2304 {(x + 4)}^{2}}}{2 \cdot - 64}$

Rewrite $2304 {(x + 4)}^{2}$ as ${(48 (x + 4))}^{2}$ .

$y = \frac{- 128 \pm \sqrt{{(48 (x + 4))}^{2}}}{2 \cdot - 64}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{- 128 \pm 48 (x + 4)}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm (48 x + 48 \cdot 4)}{2 \cdot - 64}$

Multiply $48$ by $4$ .

$y = \frac{- 128 \pm (48 x + 192)}{2 \cdot - 64}$

Multiply $2$ by $- 64$ .

$y = \frac{- 128 \pm (48 x + 192)}{- 128}$

Simplify $\frac{- 128 \pm (48 x + 192)}{- 128}$ .

$y = \frac{128 \pm (- (- 48 x - 192))}{128}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{128 \pm (- (- 48 x) + 192)}{128}$

Multiply $- 48$ by $- 1$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Multiply $- 1$ by $- 192$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $128$ to the power of $2$ .

$y = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 64 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Multiply $- 4$ by $- 64$ .

$y = \frac{- 128 \pm \sqrt{16384 + 256 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm \sqrt{16384 + 256 (9 x^{2}) + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Simplify.

Multiply $9$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $72$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $256$ by $80$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 20480}}{2 \cdot - 64}$

Add $16384$ and $20480$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 18432 x + 36864}}{2 \cdot - 64}$

Rewrite $2304 x^{2} + 18432 x + 36864$ in a factored form.

Factor $2304$ out of $2304 x^{2} + 18432 x + 36864$ .

Factor $2304$ out of $2304 x^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 18432 x + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $18432 x$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 2304 (8 x) + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $36864$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 2304 (8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 x^{2} + 2304 (8 x)$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 (x^{2} + 8 x) + 2304 \cdot 16$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 16)}}{2 \cdot - 64}$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 4^{2})}}{2 \cdot - 64}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 2 \cdot x \cdot 4 + 4^{2})}}{2 \cdot - 64}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$y = \frac{- 128 \pm \sqrt{2304 {(x + 4)}^{2}}}{2 \cdot - 64}$

Rewrite $2304 {(x + 4)}^{2}$ as ${(48 (x + 4))}^{2}$ .

$y = \frac{- 128 \pm \sqrt{{(48 (x + 4))}^{2}}}{2 \cdot - 64}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{- 128 \pm 48 (x + 4)}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm (48 x + 48 \cdot 4)}{2 \cdot - 64}$

Multiply $48$ by $4$ .

$y = \frac{- 128 \pm (48 x + 192)}{2 \cdot - 64}$

Multiply $2$ by $- 64$ .

$y = \frac{- 128 \pm (48 x + 192)}{- 128}$

Simplify $\frac{- 128 \pm (48 x + 192)}{- 128}$ .

$y = \frac{128 \pm (- (- 48 x - 192))}{128}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{128 \pm (- (- 48 x) + 192)}{128}$

Multiply $- 48$ by $- 1$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Multiply $- 1$ by $- 192$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Change the $\pm$ to $+$ .

$y = \frac{128 + 48 x + 192}{128}$

Cancel the common factor of $128 + 48 x + 192$ and $128$ .

Factor $16$ out of $128$ .

$y = \frac{16 (8) + 48 x + 192}{128}$

Factor $16$ out of $48 x$ .

$y = \frac{16 (8) + 16 (3 x) + 192}{128}$

Factor $16$ out of $192$ .

$y = \frac{16 \cdot 8 + 16 (3 x) + 16 \cdot 12}{128}$

Factor $16$ out of $16 (3 x) + 16 (12)$ .

$y = \frac{16 (8) + 16 (3 x + 12)}{128}$

Factor $16$ out of $16 (8) + 16 (3 x + 12)$ .

$y = \frac{16 (8 + 3 x + 12)}{128}$

Cancel the common factors.

Factor $16$ out of $128$ .

$y = \frac{16 (8 + 3 x + 12)}{16 \cdot 8}$

Cancel the common factor.

$y = \frac{16 (8 + 3 x + 12)}{16 \cdot 8}$

Rewrite the expression.

$y = \frac{8 + 3 x + 12}{8}$

Add $8$ and $12$ .

$y = \frac{3 x + 20}{8}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $128$ to the power of $2$ .

$y = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 64 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Multiply $- 4$ by $- 64$ .

$y = \frac{- 128 \pm \sqrt{16384 + 256 \cdot (9 x^{2} + 72 x + 80)}}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm \sqrt{16384 + 256 (9 x^{2}) + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Simplify.

Multiply $9$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 256 (72 x) + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $72$ by $256$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 256 \cdot 80}}{2 \cdot - 64}$

Multiply $256$ by $80$ .

$y = \frac{- 128 \pm \sqrt{16384 + 2304 x^{2} + 18432 x + 20480}}{2 \cdot - 64}$

Add $16384$ and $20480$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 18432 x + 36864}}{2 \cdot - 64}$

Rewrite $2304 x^{2} + 18432 x + 36864$ in a factored form.

Factor $2304$ out of $2304 x^{2} + 18432 x + 36864$ .

Factor $2304$ out of $2304 x^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 18432 x + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $18432 x$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2}) + 2304 (8 x) + 36864}}{2 \cdot - 64}$

Factor $2304$ out of $36864$ .

$y = \frac{- 128 \pm \sqrt{2304 x^{2} + 2304 (8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 x^{2} + 2304 (8 x)$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x) + 2304 \cdot 16}}{2 \cdot - 64}$

Factor $2304$ out of $2304 (x^{2} + 8 x) + 2304 \cdot 16$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 16)}}{2 \cdot - 64}$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 8 x + 4^{2})}}{2 \cdot - 64}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$y = \frac{- 128 \pm \sqrt{2304 (x^{2} + 2 \cdot x \cdot 4 + 4^{2})}}{2 \cdot - 64}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$y = \frac{- 128 \pm \sqrt{2304 {(x + 4)}^{2}}}{2 \cdot - 64}$

Rewrite $2304 {(x + 4)}^{2}$ as ${(48 (x + 4))}^{2}$ .

$y = \frac{- 128 \pm \sqrt{{(48 (x + 4))}^{2}}}{2 \cdot - 64}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{- 128 \pm 48 (x + 4)}{2 \cdot - 64}$

Apply the distributive property.

$y = \frac{- 128 \pm (48 x + 48 \cdot 4)}{2 \cdot - 64}$

Multiply $48$ by $4$ .

$y = \frac{- 128 \pm (48 x + 192)}{2 \cdot - 64}$

Multiply $2$ by $- 64$ .

$y = \frac{- 128 \pm (48 x + 192)}{- 128}$

Simplify $\frac{- 128 \pm (48 x + 192)}{- 128}$ .

$y = \frac{128 \pm (- (- 48 x - 192))}{128}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{128 \pm (- (- 48 x) + 192)}{128}$

Multiply $- 48$ by $- 1$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Multiply $- 1$ by $- 192$ .

$y = \frac{128 \pm (48 x + 192)}{128}$

Change the $\pm$ to $-$ .

$y = \frac{128 - (48 x + 192)}{128}$

Cancel the common factor of $128 - (48 x + 192)$ and $128$ .

Rewrite $128$ as $- 1 (- 128)$ .

$y = \frac{- 1 \cdot - 128 - (48 x + 192)}{128}$

Factor $- 1$ out of $- 1 (- 128) - (48 x + 192)$ .

$y = \frac{- 1 (- 128 + 48 x + 192)}{128}$

Factor $16$ out of $- 1 (- 128 + 48 x + 192)$ .

$y = \frac{16 (- 1 (- 8 + 3 x + 12))}{128}$

Cancel the common factors.

Factor $16$ out of $128$ .

$y = \frac{16 (- 1 (- 8 + 3 x + 12))}{16 (8)}$

Cancel the common factor.

$y = \frac{16 (- 1 (- 8 + 3 x + 12))}{16 \cdot 8}$

Rewrite the expression.

$y = \frac{- 1 (- 8 + 3 x + 12)}{8}$

Add $- 8$ and $12$ .

$y = \frac{- 1 (3 x + 4)}{8}$

Move the negative in front of the fraction.

$y = - \frac{3 x + 4}{8}$

The final answer is the combination of both solutions.

$y = \frac{3 x + 20}{8}$

$y = - \frac{3 x + 4}{8}$

$y = \frac{3 x + 20}{8}$

$y = - \frac{3 x + 4}{8}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Split the fraction $\frac{3 x + 20}{8}$ into two fractions.

$y = \frac{3 x}{8} + \frac{20}{8}$

$y = - \frac{3 x + 4}{8}$

Step 4

Cancel the common factor of $20$ and $8$ .

Factor $4$ out of $20$ .

$y = \frac{3 x}{8} + \frac{4 (5)}{8}$

$y = - \frac{3 x + 4}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$y = \frac{3 x}{8} + \frac{4 \cdot 5}{4 \cdot 2}$

$y = - \frac{3 x + 4}{8}$

Cancel the common factor.

$y = \frac{3 x}{8} + \frac{4 \cdot 5}{4 \cdot 2}, - \frac{3 x + 4}{8}$

Rewrite the expression.