Write in Standard Form -5x^2+60x+7y^2+84y+72=0

$- 5 x^{2} + 60 x + 7 y^{2} + 84 y + 72 = 0$

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 $a = 7$ , $b = 84$ , and $c = - 5 x^{2} + 60 x + 72$ into the quadratic formula and solve for $y$ .

$\frac{- 84 \pm \sqrt{84^{2} - 4 \cdot (7 \cdot (- 5 x^{2} + 60 x + 72))}}{2 \cdot 7}$

Simplify.

Simplify the numerator.

Raise $84$ to the power of $2$ .

$y = \frac{- 84 \pm \sqrt{7056 - 4 \cdot 7 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Multiply $- 4$ by $7$ .

$y = \frac{- 84 \pm \sqrt{7056 - 28 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm \sqrt{7056 - 28 (- 5 x^{2}) - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Simplify.

Multiply $- 5$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Multiply $60$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 28 \cdot 72}}{2 \cdot 7}$

Multiply $- 28$ by $72$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 2016}}{2 \cdot 7}$

Subtract $2016$ from $7056$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} - 1680 x + 5040}}{2 \cdot 7}$

Rewrite $140 x^{2} - 1680 x + 5040$ in a factored form.

Factor $140$ out of $140 x^{2} - 1680 x + 5040$ .

Factor $140$ out of $140 x^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) - 1680 x + 5040}}{2 \cdot 7}$

Factor $140$ out of $- 1680 x$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) + 140 (- 12 x) + 5040}}{2 \cdot 7}$

Factor $140$ out of $5040$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} + 140 (- 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 x^{2} + 140 (- 12 x)$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 (x^{2} - 12 x) + 140 \cdot 36$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 36)}}{2 \cdot 7}$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 6^{2})}}{2 \cdot 7}$

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

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 2 \cdot x \cdot 6 + 6^{2})}}{2 \cdot 7}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$y = \frac{- 84 \pm \sqrt{140 {(x - 6)}^{2}}}{2 \cdot 7}$

Rewrite $140 {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$y = \frac{- 84 \pm \sqrt{4 (35) {(x - 6)}^{2}}}{2 \cdot 7}$

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

$y = \frac{- 84 \pm \sqrt{2^{2} \cdot (35 {(x - 6)}^{2})}}{2 \cdot 7}$

Move $35$ .

$y = \frac{- 84 \pm \sqrt{2^{2} {(x - 6)}^{2} \cdot 35}}{2 \cdot 7}$

Rewrite $2^{2} {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2}$ .

$y = \frac{- 84 \pm \sqrt{{(2 (x - 6))}^{2} \cdot 35}}{2 \cdot 7}$

Pull terms out from under the radical.

$y = \frac{- 84 \pm 2 (x - 6) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x + 2 \cdot - 6) \sqrt{35}}{2 \cdot 7}$

Multiply $2$ by $- 6$ .

$y = \frac{- 84 \pm (2 x - 12) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{2 \cdot 7}$

Multiply $2$ by $7$ .

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{14}$

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

Simplify the numerator.

Raise $84$ to the power of $2$ .

$y = \frac{- 84 \pm \sqrt{7056 - 4 \cdot 7 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Multiply $- 4$ by $7$ .

$y = \frac{- 84 \pm \sqrt{7056 - 28 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm \sqrt{7056 - 28 (- 5 x^{2}) - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Simplify.

Multiply $- 5$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Multiply $60$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 28 \cdot 72}}{2 \cdot 7}$

Multiply $- 28$ by $72$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 2016}}{2 \cdot 7}$

Subtract $2016$ from $7056$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} - 1680 x + 5040}}{2 \cdot 7}$

Rewrite $140 x^{2} - 1680 x + 5040$ in a factored form.

Factor $140$ out of $140 x^{2} - 1680 x + 5040$ .

Factor $140$ out of $140 x^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) - 1680 x + 5040}}{2 \cdot 7}$

Factor $140$ out of $- 1680 x$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) + 140 (- 12 x) + 5040}}{2 \cdot 7}$

Factor $140$ out of $5040$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} + 140 (- 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 x^{2} + 140 (- 12 x)$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 (x^{2} - 12 x) + 140 \cdot 36$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 36)}}{2 \cdot 7}$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 6^{2})}}{2 \cdot 7}$

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

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 2 \cdot x \cdot 6 + 6^{2})}}{2 \cdot 7}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$y = \frac{- 84 \pm \sqrt{140 {(x - 6)}^{2}}}{2 \cdot 7}$

Rewrite $140 {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$y = \frac{- 84 \pm \sqrt{4 (35) {(x - 6)}^{2}}}{2 \cdot 7}$

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

$y = \frac{- 84 \pm \sqrt{2^{2} \cdot (35 {(x - 6)}^{2})}}{2 \cdot 7}$

Move $35$ .

$y = \frac{- 84 \pm \sqrt{2^{2} {(x - 6)}^{2} \cdot 35}}{2 \cdot 7}$

Rewrite $2^{2} {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2}$ .

$y = \frac{- 84 \pm \sqrt{{(2 (x - 6))}^{2} \cdot 35}}{2 \cdot 7}$

Pull terms out from under the radical.

$y = \frac{- 84 \pm 2 (x - 6) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x + 2 \cdot - 6) \sqrt{35}}{2 \cdot 7}$

Multiply $2$ by $- 6$ .

$y = \frac{- 84 \pm (2 x - 12) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{2 \cdot 7}$

Multiply $2$ by $7$ .

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{14}$

Change the $\pm$ to $+$ .

$y = \frac{- 84 + 2 x \sqrt{35} - 12 \sqrt{35}}{14}$

Cancel the common factor of $- 84 + 2 x \sqrt{35} - 12 \sqrt{35}$ and $14$ .

Factor $2$ out of $- 84$ .

$y = \frac{2 \cdot - 42 + 2 x \sqrt{35} - 12 \sqrt{35}}{14}$

Factor $2$ out of $2 x \sqrt{35}$ .

$y = \frac{2 \cdot - 42 + 2 (x \sqrt{35}) - 12 \sqrt{35}}{14}$

Factor $2$ out of $- 12 \sqrt{35}$ .

$y = \frac{2 \cdot - 42 + 2 (x \sqrt{35}) + 2 (- 6 \sqrt{35})}{14}$

Factor $2$ out of $2 (x \sqrt{35}) + 2 (- 6 \sqrt{35})$ .

$y = \frac{2 \cdot - 42 + 2 (x \sqrt{35} - 6 \sqrt{35})}{14}$

Factor $2$ out of $2 \cdot - 42 + 2 (x \sqrt{35} - 6 \sqrt{35})$ .

$y = \frac{2 \cdot (- 42 + x \sqrt{35} - 6 \sqrt{35})}{14}$

Cancel the common factors.

Factor $2$ out of $14$ .

$y = \frac{2 \cdot (- 42 + x \sqrt{35} - 6 \sqrt{35})}{2 (7)}$

Cancel the common factor.

$y = \frac{2 \cdot (- 42 + x \sqrt{35} - 6 \sqrt{35})}{2 \cdot 7}$

Rewrite the expression.

$y = \frac{- 42 + x \sqrt{35} - 6 \sqrt{35}}{7}$

Reorder terms.

$y = \frac{x \sqrt{35} - 42 - 6 \sqrt{35}}{7}$

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

Simplify the numerator.

Raise $84$ to the power of $2$ .

$y = \frac{- 84 \pm \sqrt{7056 - 4 \cdot 7 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Multiply $- 4$ by $7$ .

$y = \frac{- 84 \pm \sqrt{7056 - 28 \cdot (- 5 x^{2} + 60 x + 72)}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm \sqrt{7056 - 28 (- 5 x^{2}) - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Simplify.

Multiply $- 5$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 28 (60 x) - 28 \cdot 72}}{2 \cdot 7}$

Multiply $60$ by $- 28$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 28 \cdot 72}}{2 \cdot 7}$

Multiply $- 28$ by $72$ .

$y = \frac{- 84 \pm \sqrt{7056 + 140 x^{2} - 1680 x - 2016}}{2 \cdot 7}$

Subtract $2016$ from $7056$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} - 1680 x + 5040}}{2 \cdot 7}$

Rewrite $140 x^{2} - 1680 x + 5040$ in a factored form.

Factor $140$ out of $140 x^{2} - 1680 x + 5040$ .

Factor $140$ out of $140 x^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) - 1680 x + 5040}}{2 \cdot 7}$

Factor $140$ out of $- 1680 x$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2}) + 140 (- 12 x) + 5040}}{2 \cdot 7}$

Factor $140$ out of $5040$ .

$y = \frac{- 84 \pm \sqrt{140 x^{2} + 140 (- 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 x^{2} + 140 (- 12 x)$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x) + 140 \cdot 36}}{2 \cdot 7}$

Factor $140$ out of $140 (x^{2} - 12 x) + 140 \cdot 36$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 36)}}{2 \cdot 7}$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 12 x + 6^{2})}}{2 \cdot 7}$

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

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

$y = \frac{- 84 \pm \sqrt{140 (x^{2} - 2 \cdot x \cdot 6 + 6^{2})}}{2 \cdot 7}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$y = \frac{- 84 \pm \sqrt{140 {(x - 6)}^{2}}}{2 \cdot 7}$

Rewrite $140 {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$y = \frac{- 84 \pm \sqrt{4 (35) {(x - 6)}^{2}}}{2 \cdot 7}$

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

$y = \frac{- 84 \pm \sqrt{2^{2} \cdot (35 {(x - 6)}^{2})}}{2 \cdot 7}$

Move $35$ .

$y = \frac{- 84 \pm \sqrt{2^{2} {(x - 6)}^{2} \cdot 35}}{2 \cdot 7}$

Rewrite $2^{2} {(x - 6)}^{2}$ as ${(2 (x - 6))}^{2}$ .

$y = \frac{- 84 \pm \sqrt{{(2 (x - 6))}^{2} \cdot 35}}{2 \cdot 7}$

Pull terms out from under the radical.

$y = \frac{- 84 \pm 2 (x - 6) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x + 2 \cdot - 6) \sqrt{35}}{2 \cdot 7}$

Multiply $2$ by $- 6$ .

$y = \frac{- 84 \pm (2 x - 12) \sqrt{35}}{2 \cdot 7}$

Apply the distributive property.

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{2 \cdot 7}$

Multiply $2$ by $7$ .

$y = \frac{- 84 \pm (2 x \sqrt{35} - 12 \sqrt{35})}{14}$

Change the $\pm$ to $-$ .

$y = \frac{- 84 - (2 x \sqrt{35} - 12 \sqrt{35})}{14}$

Cancel the common factor of $- 84 - (2 x \sqrt{35} - 12 \sqrt{35})$ and $14$ .

Rewrite $- 84$ as $- 1 (84)$ .

$y = \frac{- 1 \cdot 84 - (2 x \sqrt{35} - 12 \sqrt{35})}{14}$

Factor $- 1$ out of $- 1 (84) - (2 x \sqrt{35} - 12 \sqrt{35})$ .

$y = \frac{- 1 (84 + 2 x \sqrt{35} - 12 \sqrt{35})}{14}$

Factor $2$ out of $- 1 (84 + 2 x \sqrt{35} - 12 \sqrt{35})$ .

$y = \frac{2 (- 1 (42 + x \sqrt{35} - 6 \sqrt{35}))}{14}$

Cancel the common factors.

Factor $2$ out of $14$ .

$y = \frac{2 (- 1 (42 + x \sqrt{35} - 6 \sqrt{35}))}{2 (7)}$

Cancel the common factor.

$y = \frac{2 (- 1 (42 + x \sqrt{35} - 6 \sqrt{35}))}{2 \cdot 7}$

Rewrite the expression.

$y = \frac{- 1 (42 + x \sqrt{35} - 6 \sqrt{35})}{7}$

Reorder terms.

$y = \frac{- 1 (x \sqrt{35} + 42 - 6 \sqrt{35})}{7}$

Move the negative in front of the fraction.

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

The final answer is the combination of both solutions.

$y = \frac{x \sqrt{35} - 42 - 6 \sqrt{35}}{7}$

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

$y = \frac{x \sqrt{35} - 42 - 6 \sqrt{35}}{7}$

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

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

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

Split the fraction $\frac{x \sqrt{35} - 42 - 6 \sqrt{35}}{7}$ into two fractions.

$y = \frac{x \sqrt{35} - 42}{7} + \frac{- 6 \sqrt{35}}{7}$

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

Simplify each term.

Split the fraction $\frac{x \sqrt{35} - 42}{7}$ into two fractions.

$y = \frac{x \sqrt{35}}{7} + \frac{- 42}{7} + \frac{- 6 \sqrt{35}}{7}$

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

Divide $- 42$ by $7$ .

$y = \frac{x \sqrt{35}}{7} - 6 + \frac{- 6 \sqrt{35}}{7}$

$y = - \frac{x \sqrt{35} + 42 - 6 \sqrt{35}}{7}$

Move the negative in front of the fraction.