Calculus Examples

$f (x) = x^{2} - 8 x y + y^{2} + 180 y + 46$

Step 1

Rewrite the equation as $x^{2} - 8 x y + y^{2} + 180 y + 46 = x$ .

$x^{2} - 8 x y + y^{2} + 180 y + 46 = x$

Step 2

Subtract $x$ from both sides of the equation.

$x^{2} - 8 x y + y^{2} + 180 y + 46 - x = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 8 x + 180$ , and $c = x^{2} + 46 - x$ into the quadratic formula and solve for $y$ .

$\frac{- (- 8 x + 180) \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot (1 \cdot (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$y = \frac{- (- 8 x) - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 8$ by $- 1$ .

$y = \frac{8 x - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 5.1.3

Multiply $- 1$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 5.1.4

Add parentheses.

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot (1 \cdot (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 5.1.5

Let $u = 1 \cdot (x^{2} + 46 - x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 46 - x)$ .

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Step 5.1.5.1

Rewrite ${(- 8 x + 180)}^{2}$ as $(- 8 x + 180) (- 8 x + 180)$ .

$y = \frac{8 x - 180 \pm \sqrt{(- 8 x + 180) (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.2

Expand $(- 8 x + 180) (- 8 x + 180)$ using the FOIL Method.

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Step 5.1.5.2.1

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x + 180) + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.2.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.2.3

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3

Simplify and combine like terms.

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Step 5.1.5.3.1

Simplify each term.

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Step 5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x \cdot x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.5.3.1.2.1

Move $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 (x \cdot x)) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x^{2}) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.3

Multiply $- 8$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.4

Multiply $180$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.5

Multiply $- 8$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.1.6

Multiply $180$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.5.3.2

Subtract $1440 x$ from $- 1440 x$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 5.1.6

Factor $4$ out of $64 x^{2} - 2880 x + 32400 - 4 u$ .

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Step 5.1.6.1

Factor $4$ out of $64 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 5.1.6.2

Factor $4$ out of $- 2880 x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 32400 - 4 u}}{2 \cdot 1}$

Step 5.1.6.3

Factor $4$ out of $32400$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) - 4 u}}{2 \cdot 1}$

Step 5.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 5.1.6.5

Factor $4$ out of $4 (16 x^{2}) + 4 (- 720 x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 5.1.6.6

Factor $4$ out of $4 (16 x^{2} - 720 x) + 4 (8100)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100) + 4 (- u)}}{2 \cdot 1}$

Step 5.1.6.7

Factor $4$ out of $4 (16 x^{2} - 720 x + 8100) + 4 (- u)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - u)}}{2 \cdot 1}$

Step 5.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 46 - x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (1 \cdot (x^{2} + 46 - x)))}}{2 \cdot 1}$

Step 5.1.8

Simplify.

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Step 5.1.8.1

Simplify each term.

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Step 5.1.8.1.1

Multiply $x^{2} + 46 - x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 5.1.8.1.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 1 \cdot 46 + x)}}{2 \cdot 1}$

Step 5.1.8.1.3

Simplify.

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Step 5.1.8.1.3.1

Multiply $- 1$ by $46$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 5.1.8.1.3.2

Multiply $- - x$ .

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Step 5.1.8.1.3.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + 1 x)}}{2 \cdot 1}$

Step 5.1.8.1.3.2.2

Multiply $x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 5.1.8.2

Subtract $x^{2}$ from $16 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 720 x + 8100 - 46 + x)}}{2 \cdot 1}$

Step 5.1.8.3

Add $- 720 x$ and $x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8100 - 46)}}{2 \cdot 1}$

Step 5.1.8.4

Subtract $46$ from $8100$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 5.1.9

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

$y = \frac{8 x - 180 \pm \sqrt{2^{2} (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 5.1.10

Pull terms out from under the radical.

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$y = \frac{- (- 8 x) - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 8$ by $- 1$ .

$y = \frac{8 x - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 6.1.3

Multiply $- 1$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 6.1.4

Add parentheses.

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot (1 \cdot (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 6.1.5

Let $u = 1 \cdot (x^{2} + 46 - x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 46 - x)$ .

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Step 6.1.5.1

Rewrite ${(- 8 x + 180)}^{2}$ as $(- 8 x + 180) (- 8 x + 180)$ .

$y = \frac{8 x - 180 \pm \sqrt{(- 8 x + 180) (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2

Expand $(- 8 x + 180) (- 8 x + 180)$ using the FOIL Method.

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Step 6.1.5.2.1

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x + 180) + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2.3

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3

Simplify and combine like terms.

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Step 6.1.5.3.1

Simplify each term.

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Step 6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x \cdot x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.5.3.1.2.1

Move $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 (x \cdot x)) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x^{2}) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.3

Multiply $- 8$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.4

Multiply $180$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.5

Multiply $- 8$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.6

Multiply $180$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.2

Subtract $1440 x$ from $- 1440 x$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 6.1.6

Factor $4$ out of $64 x^{2} - 2880 x + 32400 - 4 u$ .

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Step 6.1.6.1

Factor $4$ out of $64 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 6.1.6.2

Factor $4$ out of $- 2880 x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 32400 - 4 u}}{2 \cdot 1}$

Step 6.1.6.3

Factor $4$ out of $32400$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) - 4 u}}{2 \cdot 1}$

Step 6.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.5

Factor $4$ out of $4 (16 x^{2}) + 4 (- 720 x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.6

Factor $4$ out of $4 (16 x^{2} - 720 x) + 4 (8100)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.7

Factor $4$ out of $4 (16 x^{2} - 720 x + 8100) + 4 (- u)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - u)}}{2 \cdot 1}$

Step 6.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 46 - x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (1 \cdot (x^{2} + 46 - x)))}}{2 \cdot 1}$

Step 6.1.8

Simplify.

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Step 6.1.8.1

Simplify each term.

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Step 6.1.8.1.1

Multiply $x^{2} + 46 - x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 6.1.8.1.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 1 \cdot 46 + x)}}{2 \cdot 1}$

Step 6.1.8.1.3

Simplify.

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Step 6.1.8.1.3.1

Multiply $- 1$ by $46$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 6.1.8.1.3.2

Multiply $- - x$ .

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Step 6.1.8.1.3.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + 1 x)}}{2 \cdot 1}$

Step 6.1.8.1.3.2.2

Multiply $x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 6.1.8.2

Subtract $x^{2}$ from $16 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 720 x + 8100 - 46 + x)}}{2 \cdot 1}$

Step 6.1.8.3

Add $- 720 x$ and $x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8100 - 46)}}{2 \cdot 1}$

Step 6.1.8.4

Subtract $46$ from $8100$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 6.1.9

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

$y = \frac{8 x - 180 \pm \sqrt{2^{2} (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 6.1.10

Pull terms out from under the radical.

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6.3

Change the $\pm$ to $+$ .

$y = \frac{8 x - 180 + 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6.4

Cancel the common factor of $8 x - 180 + 2 \sqrt{15 x^{2} - 719 x + 8054}$ and $2$ .

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Step 6.4.1

Factor $2$ out of $8 x$ .

$y = \frac{2 (4 x) - 180 + 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6.4.2

Factor $2$ out of $- 180$ .

$y = \frac{2 (4 x) + 2 \cdot - 90 + 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6.4.3

Factor $2$ out of $2 (4 x) + 2 (- 90)$ .

$y = \frac{2 (4 x - 90) + 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 6.4.4

Factor $2$ out of $2 \sqrt{15 x^{2} - 719 x + 8054}$ .

$y = \frac{2 (4 x - 90) + 2 (\sqrt{15 x^{2} - 719 x + 8054})}{2}$

Step 6.4.5

Factor $2$ out of $2 (4 x - 90) + 2 (\sqrt{15 x^{2} - 719 x + 8054})$ .

$y = \frac{2 (4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054})}{2}$

Step 6.4.6

Cancel the common factors.

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Step 6.4.6.1

Factor $2$ out of $2$ .

$y = \frac{2 (4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054})}{2 (1)}$

Step 6.4.6.2

Cancel the common factor.

$y = \frac{2 (4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054})}{2 \cdot 1}$

Step 6.4.6.3

Rewrite the expression.

$y = \frac{4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054}}{1}$

Step 6.4.6.4

Divide $4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054}$ by $1$ .

$y = 4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$y = \frac{- (- 8 x) - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 8$ by $- 1$ .

$y = \frac{8 x - 1 \cdot 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 7.1.3

Multiply $- 1$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot 1 \cdot (x^{2} + 46 - x)}}{2 \cdot 1}$

Step 7.1.4

Add parentheses.

$y = \frac{8 x - 180 \pm \sqrt{{(- 8 x + 180)}^{2} - 4 \cdot (1 \cdot (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 7.1.5

Let $u = 1 \cdot (x^{2} + 46 - x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 46 - x)$ .

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Step 7.1.5.1

Rewrite ${(- 8 x + 180)}^{2}$ as $(- 8 x + 180) (- 8 x + 180)$ .

$y = \frac{8 x - 180 \pm \sqrt{(- 8 x + 180) (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.2

Expand $(- 8 x + 180) (- 8 x + 180)$ using the FOIL Method.

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Step 7.1.5.2.1

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x + 180) + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.2.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x + 180) - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.2.3

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{- 8 x (- 8 x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3

Simplify and combine like terms.

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Step 7.1.5.3.1

Simplify each term.

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Step 7.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x \cdot x) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1.5.3.1.2.1

Move $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 (x \cdot x)) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{8 x - 180 \pm \sqrt{- 8 \cdot (- 8 x^{2}) - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.3

Multiply $- 8$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 8 x \cdot 180 + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.4

Multiply $180$ by $- 8$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x + 180 (- 8 x) + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.5

Multiply $- 8$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 180 \cdot 180 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.1.6

Multiply $180$ by $180$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 1440 x - 1440 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

Step 7.1.5.3.2

Subtract $1440 x$ from $- 1440 x$ .

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{8 x - 180 \pm \sqrt{64 x^{2} - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 7.1.6

Factor $4$ out of $64 x^{2} - 2880 x + 32400 - 4 u$ .

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Step 7.1.6.1

Factor $4$ out of $64 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) - 2880 x + 32400 - 4 u}}{2 \cdot 1}$

Step 7.1.6.2

Factor $4$ out of $- 2880 x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 32400 - 4 u}}{2 \cdot 1}$

Step 7.1.6.3

Factor $4$ out of $32400$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) - 4 u}}{2 \cdot 1}$

Step 7.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2}) + 4 (- 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 7.1.6.5

Factor $4$ out of $4 (16 x^{2}) + 4 (- 720 x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x) + 4 (8100) + 4 (- u)}}{2 \cdot 1}$

Step 7.1.6.6

Factor $4$ out of $4 (16 x^{2} - 720 x) + 4 (8100)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100) + 4 (- u)}}{2 \cdot 1}$

Step 7.1.6.7

Factor $4$ out of $4 (16 x^{2} - 720 x + 8100) + 4 (- u)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - u)}}{2 \cdot 1}$

Step 7.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 46 - x)$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (1 \cdot (x^{2} + 46 - x)))}}{2 \cdot 1}$

Step 7.1.8

Simplify.

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Step 7.1.8.1

Simplify each term.

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Step 7.1.8.1.1

Multiply $x^{2} + 46 - x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - (x^{2} + 46 - x))}}{2 \cdot 1}$

Step 7.1.8.1.2

Apply the distributive property.

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 1 \cdot 46 + x)}}{2 \cdot 1}$

Step 7.1.8.1.3

Simplify.

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Step 7.1.8.1.3.1

Multiply $- 1$ by $46$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 7.1.8.1.3.2

Multiply $- - x$ .

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Step 7.1.8.1.3.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + 1 x)}}{2 \cdot 1}$

Step 7.1.8.1.3.2.2

Multiply $x$ by $1$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (16 x^{2} - 720 x + 8100 - x^{2} - 46 + x)}}{2 \cdot 1}$

Step 7.1.8.2

Subtract $x^{2}$ from $16 x^{2}$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 720 x + 8100 - 46 + x)}}{2 \cdot 1}$

Step 7.1.8.3

Add $- 720 x$ and $x$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8100 - 46)}}{2 \cdot 1}$

Step 7.1.8.4

Subtract $46$ from $8100$ .

$y = \frac{8 x - 180 \pm \sqrt{4 (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 7.1.9

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

$y = \frac{8 x - 180 \pm \sqrt{2^{2} (15 x^{2} - 719 x + 8054)}}{2 \cdot 1}$

Step 7.1.10

Pull terms out from under the radical.

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$y = \frac{8 x - 180 \pm 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 7.3

Change the $\pm$ to $-$ .

$y = \frac{8 x - 180 - 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 7.4

Cancel the common factor of $8 x - 180 - 2 \sqrt{15 x^{2} - 719 x + 8054}$ and $2$ .

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Step 7.4.1

Factor $2$ out of $8 x$ .

$y = \frac{2 (4 x) - 180 - 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 7.4.2

Factor $2$ out of $- 180$ .

$y = \frac{2 (4 x) + 2 \cdot - 90 - 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 7.4.3

Factor $2$ out of $2 (4 x) + 2 (- 90)$ .

$y = \frac{2 (4 x - 90) - 2 \sqrt{15 x^{2} - 719 x + 8054}}{2}$

Step 7.4.4

Factor $2$ out of $- 2 \sqrt{15 x^{2} - 719 x + 8054}$ .

$y = \frac{2 (4 x - 90) + 2 (- \sqrt{15 x^{2} - 719 x + 8054})}{2}$

Step 7.4.5

Factor $2$ out of $2 (4 x - 90) + 2 (- \sqrt{15 x^{2} - 719 x + 8054})$ .

$y = \frac{2 (4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054})}{2}$

Step 7.4.6

Cancel the common factors.

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Step 7.4.6.1

Factor $2$ out of $2$ .

$y = \frac{2 (4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054})}{2 (1)}$

Step 7.4.6.2

Cancel the common factor.

$y = \frac{2 (4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054})}{2 \cdot 1}$

Step 7.4.6.3

Rewrite the expression.

$y = \frac{4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054}}{1}$

Step 7.4.6.4

Divide $4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054}$ by $1$ .

$y = 4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054}$

Step 8

The final answer is the combination of both solutions.

$y = 4 x - 90 + \sqrt{15 x^{2} - 719 x + 8054}$

$y = 4 x - 90 - \sqrt{15 x^{2} - 719 x + 8054}$

Step 9

Set the radicand in $\sqrt{15 x^{2} - 719 x + 8054}$ greater than or equal to $0$ to find where the expression is defined.

$15 x^{2} - 719 x + 8054 \geq 0$

Step 10

Solve for $x$ .

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Step 10.1

Convert the inequality to an equation.

$15 x^{2} - 719 x + 8054 = 0$

Step 10.2

Use the quadratic formula to find the solutions.

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

Step 10.3

Substitute the values $a = 15$ , $b = - 719$ , and $c = 8054$ into the quadratic formula and solve for $x$ .

$\frac{719 \pm \sqrt{{(- 719)}^{2} - 4 \cdot (15 \cdot 8054)}}{2 \cdot 15}$

Step 10.4

Simplify.

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Step 10.4.1

Simplify the numerator.

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Step 10.4.1.1

Raise $- 719$ to the power of $2$ .

$x = \frac{719 \pm \sqrt{516961 - 4 \cdot 15 \cdot 8054}}{2 \cdot 15}$

Step 10.4.1.2

Multiply $- 4 \cdot 15 \cdot 8054$ .

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Step 10.4.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{719 \pm \sqrt{516961 - 60 \cdot 8054}}{2 \cdot 15}$

Step 10.4.1.2.2

Multiply $- 60$ by $8054$ .

$x = \frac{719 \pm \sqrt{516961 - 483240}}{2 \cdot 15}$

Step 10.4.1.3

Subtract $483240$ from $516961$ .

$x = \frac{719 \pm \sqrt{33721}}{2 \cdot 15}$

Step 10.4.2

Multiply $2$ by $15$ .

$x = \frac{719 \pm \sqrt{33721}}{30}$

Step 10.5

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

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Step 10.5.1

Simplify the numerator.

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Step 10.5.1.1

Raise $- 719$ to the power of $2$ .

$x = \frac{719 \pm \sqrt{516961 - 4 \cdot 15 \cdot 8054}}{2 \cdot 15}$

Step 10.5.1.2

Multiply $- 4 \cdot 15 \cdot 8054$ .

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Step 10.5.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{719 \pm \sqrt{516961 - 60 \cdot 8054}}{2 \cdot 15}$

Step 10.5.1.2.2

Multiply $- 60$ by $8054$ .

$x = \frac{719 \pm \sqrt{516961 - 483240}}{2 \cdot 15}$

Step 10.5.1.3

Subtract $483240$ from $516961$ .

$x = \frac{719 \pm \sqrt{33721}}{2 \cdot 15}$

Step 10.5.2

Multiply $2$ by $15$ .

$x = \frac{719 \pm \sqrt{33721}}{30}$

Step 10.5.3

Change the $\pm$ to $+$ .

$x = \frac{719 + \sqrt{33721}}{30}$

Step 10.6

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

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Step 10.6.1

Simplify the numerator.

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Step 10.6.1.1

Raise $- 719$ to the power of $2$ .

$x = \frac{719 \pm \sqrt{516961 - 4 \cdot 15 \cdot 8054}}{2 \cdot 15}$

Step 10.6.1.2

Multiply $- 4 \cdot 15 \cdot 8054$ .

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Step 10.6.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{719 \pm \sqrt{516961 - 60 \cdot 8054}}{2 \cdot 15}$

Step 10.6.1.2.2

Multiply $- 60$ by $8054$ .

$x = \frac{719 \pm \sqrt{516961 - 483240}}{2 \cdot 15}$

Step 10.6.1.3

Subtract $483240$ from $516961$ .

$x = \frac{719 \pm \sqrt{33721}}{2 \cdot 15}$

Step 10.6.2

Multiply $2$ by $15$ .

$x = \frac{719 \pm \sqrt{33721}}{30}$

Step 10.6.3

Change the $\pm$ to $-$ .

$x = \frac{719 - \sqrt{33721}}{30}$

Step 10.7

Consolidate the solutions.

$x = \frac{719 + \sqrt{33721}}{30}, \frac{719 - \sqrt{33721}}{30}$

Step 10.8

Use each root to create test intervals.

$x < \frac{719 - \sqrt{33721}}{30}$

$\frac{719 - \sqrt{33721}}{30} < x < \frac{719 + \sqrt{33721}}{30}$

$x > \frac{719 + \sqrt{33721}}{30}$

Step 10.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.9.1

Test a value on the interval $x < \frac{719 - \sqrt{33721}}{30}$ to see if it makes the inequality true.

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Step 10.9.1.1

Choose a value on the interval $x < \frac{719 - \sqrt{33721}}{30}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.9.1.2

Replace $x$ with $0$ in the original inequality.

$15 {(0)}^{2} - 719 \cdot 0 + 8054 \geq 0$

Step 10.9.1.3

The left side $8054$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.9.2

Test a value on the interval $\frac{719 - \sqrt{33721}}{30} < x < \frac{719 + \sqrt{33721}}{30}$ to see if it makes the inequality true.

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Step 10.9.2.1

Choose a value on the interval $\frac{719 - \sqrt{33721}}{30} < x < \frac{719 + \sqrt{33721}}{30}$ and see if this value makes the original inequality true.

$x = 20$

Step 10.9.2.2

Replace $x$ with $20$ in the original inequality.

$15 {(20)}^{2} - 719 \cdot 20 + 8054 \geq 0$

Step 10.9.2.3

The left side $- 326$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.9.3

Test a value on the interval $x > \frac{719 + \sqrt{33721}}{30}$ to see if it makes the inequality true.

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Step 10.9.3.1

Choose a value on the interval $x > \frac{719 + \sqrt{33721}}{30}$ and see if this value makes the original inequality true.

$x = 33$

Step 10.9.3.2

Replace $x$ with $33$ in the original inequality.

$15 {(33)}^{2} - 719 \cdot 33 + 8054 \geq 0$

Step 10.9.3.3

The left side $662$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{719 - \sqrt{33721}}{30}$ True

$\frac{719 - \sqrt{33721}}{30} < x < \frac{719 + \sqrt{33721}}{30}$ False

$x > \frac{719 + \sqrt{33721}}{30}$ True

$x < \frac{719 - \sqrt{33721}}{30}$ True

$\frac{719 - \sqrt{33721}}{30} < x < \frac{719 + \sqrt{33721}}{30}$ False

$x > \frac{719 + \sqrt{33721}}{30}$ True

Step 10.10

The solution consists of all of the true intervals.

$x \leq \frac{719 - \sqrt{33721}}{30}$ or $x \geq \frac{719 + \sqrt{33721}}{30}$

Step 11

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, \frac{719 - \sqrt{33721}}{30}] \cup [\frac{719 + \sqrt{33721}}{30}, \infty)$

Set-Builder Notation:

${x | x \leq \frac{719 - \sqrt{33721}}{30}, x \geq \frac{719 + \sqrt{33721}}{30}}$

Step 12

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \frac{176 - \sqrt{33721}}{30}] \cup [\frac{176 + \sqrt{33721}}{30}, \infty)$

Set-Builder Notation:

${y | y \leq \frac{176 - \sqrt{33721}}{30}, y \geq \frac{176 + \sqrt{33721}}{30}}$

Step 13

Determine the domain and range.

Domain: $(- \infty, \frac{719 - \sqrt{33721}}{30}] \cup [\frac{719 + \sqrt{33721}}{30}, \infty), {x | x \leq \frac{719 - \sqrt{33721}}{30}, x \geq \frac{719 + \sqrt{33721}}{30}}$

Range: $(- \infty, \frac{176 - \sqrt{33721}}{30}] \cup [\frac{176 + \sqrt{33721}}{30}, \infty), {y | y \leq \frac{176 - \sqrt{33721}}{30}, y \geq \frac{176 + \sqrt{33721}}{30}}$

Step 14