Algebra Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

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

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

Step 1.3.1.2

Multiply $- 4$ by $7$ .

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

Step 1.3.1.3

Apply the distributive property.

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

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $- 5$ by $- 28$ .

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

Step 1.3.1.4.2

Multiply $60$ by $- 28$ .

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

Step 1.3.1.4.3

Multiply $- 28$ by $72$ .

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

Step 1.3.1.5

Subtract $2016$ from $7056$ .

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

Step 1.3.1.6

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

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Step 1.3.1.6.1

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

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Step 1.3.1.6.1.1

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

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

Step 1.3.1.6.1.2

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

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

Step 1.3.1.6.1.3

Factor $140$ out of $5040$ .

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

Step 1.3.1.6.1.4

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

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

Step 1.3.1.6.1.5

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

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

Step 1.3.1.6.2

Factor using the perfect square rule.

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Step 1.3.1.6.2.1

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

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

Step 1.3.1.6.2.2

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

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

Step 1.3.1.6.2.3

Rewrite the polynomial.

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

Step 1.3.1.6.2.4

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

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

Step 1.3.1.7

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

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Step 1.3.1.7.1

Factor $4$ out of $140$ .

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

Step 1.3.1.7.2

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

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

Step 1.3.1.7.3

Move $35$ .

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

Step 1.3.1.7.4

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

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

Step 1.3.1.8

Pull terms out from under the radical.

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

Step 1.3.1.9

Apply the distributive property.

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

Step 1.3.1.10

Multiply $2$ by $- 6$ .

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

Step 1.3.1.11

Apply the distributive property.

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

Step 1.3.2

Multiply $2$ by $7$ .

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

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

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

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

Step 1.4.1.2

Multiply $- 4$ by $7$ .

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

Step 1.4.1.3

Apply the distributive property.

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

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $- 5$ by $- 28$ .

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

Step 1.4.1.4.2

Multiply $60$ by $- 28$ .

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

Step 1.4.1.4.3

Multiply $- 28$ by $72$ .

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

Step 1.4.1.5

Subtract $2016$ from $7056$ .

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

Step 1.4.1.6

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

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Step 1.4.1.6.1

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

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Step 1.4.1.6.1.1

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

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

Step 1.4.1.6.1.2

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

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

Step 1.4.1.6.1.3

Factor $140$ out of $5040$ .

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

Step 1.4.1.6.1.4

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

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

Step 1.4.1.6.1.5

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

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

Step 1.4.1.6.2

Factor using the perfect square rule.

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Step 1.4.1.6.2.1

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

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

Step 1.4.1.6.2.2

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

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

Step 1.4.1.6.2.3

Rewrite the polynomial.

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

Step 1.4.1.6.2.4

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

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

Step 1.4.1.7

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

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Step 1.4.1.7.1

Factor $4$ out of $140$ .

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

Step 1.4.1.7.2

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

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

Step 1.4.1.7.3

Move $35$ .

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

Step 1.4.1.7.4

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

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

Step 1.4.1.8

Pull terms out from under the radical.

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

Step 1.4.1.9

Apply the distributive property.

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

Step 1.4.1.10

Multiply $2$ by $- 6$ .

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

Step 1.4.1.11

Apply the distributive property.

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

Step 1.4.2

Multiply $2$ by $7$ .

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

Step 1.4.3

Change the $\pm$ to $+$ .

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

Step 1.4.4

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

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Step 1.4.4.1

Factor $2$ out of $- 84$ .

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

Step 1.4.4.2

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

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

Step 1.4.4.3

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

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

Step 1.4.4.4

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

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

Step 1.4.4.5

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

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

Step 1.4.4.6

Cancel the common factors.

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Step 1.4.4.6.1

Factor $2$ out of $14$ .

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

Step 1.4.4.6.2

Cancel the common factor.

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

Step 1.4.4.6.3

Rewrite the expression.

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

Step 1.4.5

Reorder terms.

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

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

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

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

Step 1.5.1.2

Multiply $- 4$ by $7$ .

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

Step 1.5.1.3

Apply the distributive property.

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

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $- 5$ by $- 28$ .

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

Step 1.5.1.4.2

Multiply $60$ by $- 28$ .

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

Step 1.5.1.4.3

Multiply $- 28$ by $72$ .

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

Step 1.5.1.5

Subtract $2016$ from $7056$ .

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

Step 1.5.1.6

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

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Step 1.5.1.6.1

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

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Step 1.5.1.6.1.1

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

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

Step 1.5.1.6.1.2

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

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

Step 1.5.1.6.1.3

Factor $140$ out of $5040$ .

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

Step 1.5.1.6.1.4

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

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

Step 1.5.1.6.1.5

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

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

Step 1.5.1.6.2

Factor using the perfect square rule.

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Step 1.5.1.6.2.1

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

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

Step 1.5.1.6.2.2

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

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

Step 1.5.1.6.2.3

Rewrite the polynomial.

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

Step 1.5.1.6.2.4

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

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

Step 1.5.1.7

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

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Step 1.5.1.7.1

Factor $4$ out of $140$ .

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

Step 1.5.1.7.2

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

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

Step 1.5.1.7.3

Move $35$ .

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

Step 1.5.1.7.4

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

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

Step 1.5.1.8

Pull terms out from under the radical.

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

Step 1.5.1.9

Apply the distributive property.

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

Step 1.5.1.10

Multiply $2$ by $- 6$ .

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

Step 1.5.1.11

Apply the distributive property.

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

Step 1.5.2

Multiply $2$ by $7$ .

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

Step 1.5.3

Change the $\pm$ to $-$ .

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

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

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

Step 1.5.4.3

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

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

Step 1.5.4.4

Cancel the common factors.

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Step 1.5.4.4.1

Factor $2$ out of $14$ .

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

Step 1.5.4.4.2

Cancel the common factor.

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

Step 1.5.4.4.3

Rewrite the expression.

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

Step 1.5.5

Reorder terms.

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

Step 1.5.6

Move the negative in front of the fraction.

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

Step 1.6

The final answer is the combination of both solutions.

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

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

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

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

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

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

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

Step 4

Simplify each term.

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Step 4.1

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

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

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

Step 4.2

Divide $- 42$ by $7$ .

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

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

Step 4.3

Move the negative in front of the fraction.

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

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

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

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

Step 5

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

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

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

Step 6.2

Divide $42$ by $7$ .

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

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

Step 6.3

Move the negative in front of the fraction.

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

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

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

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

Step 7

Apply the distributive property.

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

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

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $6$ .

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

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

Step 8.2

Multiply $- - \frac{6 \sqrt{35}}{7}$ .

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Step 8.2.1

Multiply $- 1$ by $- 1$ .

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

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

Step 8.2.2

Multiply $\frac{6 \sqrt{35}}{7}$ by $1$ .

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

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

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

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

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

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

Step 9

Reorder terms.

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

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

Step 10

Remove parentheses.

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

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

Step 11