Calculus Examples

$f (x) = - 2 x^{2} - x y - 3 y^{2} + 50 x + 70 y - 200$

Step 1

Rewrite the equation as $- 2 x^{2} - x y - 3 y^{2} + 50 x + 70 y - 200 = x$ .

$- 2 x^{2} - x y - 3 y^{2} + 50 x + 70 y - 200 = x$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 2 x^{2} - x y - 3 y^{2} + 50 x + 70 y - 200 - x = 0$

Step 2.2

Subtract $x$ from $50 x$ .

$- 2 x^{2} - x y - 3 y^{2} + 49 x + 70 y - 200 = 0$

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 $a = - 3$ , $b = - x + 70$ , and $c = - 2 x^{2} + 49 x - 200$ into the quadratic formula and solve for $y$ .

$\frac{- (- x + 70) \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot (- 3 \cdot (- 2 x^{2} + 49 x - 200))}}{2 \cdot - 3}$

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.

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.2

Multiply $- - x$ .

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Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1 x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.2.2

Multiply $x$ by $1$ .

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.3

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.4

Rewrite ${(- x + 70)}^{2}$ as $(- x + 70) (- x + 70)$ .

$y = \frac{x - 70 \pm \sqrt{(- x + 70) (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.5

Expand $(- x + 70) (- x + 70)$ using the FOIL Method.

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Step 5.1.5.1

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x + 70) + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.5.2

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.5.3

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6

Simplify and combine like terms.

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Step 5.1.6.1

Simplify each term.

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Step 5.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x \cdot x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.2

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

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Step 5.1.6.1.2.1

Move $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 (x \cdot x)) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x^{2}) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.3

Multiply $- 1$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{1 x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.4

Multiply $x^{2}$ by $1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.5

Multiply $70$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.6

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.1.7

Multiply $70$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.6.2

Subtract $70 x$ from $- 70 x$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.7

Multiply $- 4$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 5.1.8

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 (- 2 x^{2}) + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 5.1.9

Simplify.

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Step 5.1.9.1

Multiply $- 2$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 5.1.9.2

Multiply $49$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x + 12 \cdot - 200}}{2 \cdot - 3}$

Step 5.1.9.3

Multiply $12$ by $- 200$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x - 2400}}{2 \cdot - 3}$

Step 5.1.10

Subtract $24 x^{2}$ from $x^{2}$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} - 140 x + 4900 + 588 x - 2400}}{2 \cdot - 3}$

Step 5.1.11

Add $- 140 x$ and $588 x$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 4900 - 2400}}{2 \cdot - 3}$

Step 5.1.12

Subtract $2400$ from $4900$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{2 \cdot - 3}$

Step 5.2

Multiply $2$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{- 6}$

Step 5.3

Move the negative in front of the fraction.

$y = - \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

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.

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- - x$ .

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1 x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.2.2

Multiply $x$ by $1$ .

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.3

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.4

Rewrite ${(- x + 70)}^{2}$ as $(- x + 70) (- x + 70)$ .

$y = \frac{x - 70 \pm \sqrt{(- x + 70) (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.5

Expand $(- x + 70) (- x + 70)$ using the FOIL Method.

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Step 6.1.5.1

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x + 70) + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.5.2

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.5.3

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6

Simplify and combine like terms.

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Step 6.1.6.1

Simplify each term.

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Step 6.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x \cdot x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.2

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

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Step 6.1.6.1.2.1

Move $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 (x \cdot x)) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x^{2}) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.3

Multiply $- 1$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{1 x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.4

Multiply $x^{2}$ by $1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.5

Multiply $70$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.6

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.1.7

Multiply $70$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.6.2

Subtract $70 x$ from $- 70 x$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.7

Multiply $- 4$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 6.1.8

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 (- 2 x^{2}) + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 6.1.9

Simplify.

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Step 6.1.9.1

Multiply $- 2$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 6.1.9.2

Multiply $49$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x + 12 \cdot - 200}}{2 \cdot - 3}$

Step 6.1.9.3

Multiply $12$ by $- 200$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x - 2400}}{2 \cdot - 3}$

Step 6.1.10

Subtract $24 x^{2}$ from $x^{2}$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} - 140 x + 4900 + 588 x - 2400}}{2 \cdot - 3}$

Step 6.1.11

Add $- 140 x$ and $588 x$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 4900 - 2400}}{2 \cdot - 3}$

Step 6.1.12

Subtract $2400$ from $4900$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{- 6}$

Step 6.3

Move the negative in front of the fraction.

$y = - \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

Step 6.4

Change the $\pm$ to $+$ .

$y = - \frac{x - 70 + \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

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.

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.2

Multiply $- - x$ .

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Step 7.1.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1 x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.2.2

Multiply $x$ by $1$ .

$y = \frac{x - 1 \cdot 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.3

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{{(- x + 70)}^{2} - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.4

Rewrite ${(- x + 70)}^{2}$ as $(- x + 70) (- x + 70)$ .

$y = \frac{x - 70 \pm \sqrt{(- x + 70) (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.5

Expand $(- x + 70) (- x + 70)$ using the FOIL Method.

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Step 7.1.5.1

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x + 70) + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.5.2

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x + 70) - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.5.3

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{- x (- x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6

Simplify and combine like terms.

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Step 7.1.6.1

Simplify each term.

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Step 7.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x \cdot x) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.2

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

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Step 7.1.6.1.2.1

Move $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 (x \cdot x)) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{x - 70 \pm \sqrt{- 1 \cdot (- 1 x^{2}) - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.3

Multiply $- 1$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{1 x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.4

Multiply $x^{2}$ by $1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - x \cdot 70 + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.5

Multiply $70$ by $- 1$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x + 70 (- x) + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.6

Multiply $- 1$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 70 \cdot 70 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.1.7

Multiply $70$ by $70$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 70 x - 70 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.6.2

Subtract $70 x$ from $- 70 x$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 4 \cdot - 3 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.7

Multiply $- 4$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 \cdot (- 2 x^{2} + 49 x - 200)}}{2 \cdot - 3}$

Step 7.1.8

Apply the distributive property.

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 + 12 (- 2 x^{2}) + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 7.1.9

Simplify.

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Step 7.1.9.1

Multiply $- 2$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 12 (49 x) + 12 \cdot - 200}}{2 \cdot - 3}$

Step 7.1.9.2

Multiply $49$ by $12$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x + 12 \cdot - 200}}{2 \cdot - 3}$

Step 7.1.9.3

Multiply $12$ by $- 200$ .

$y = \frac{x - 70 \pm \sqrt{x^{2} - 140 x + 4900 - 24 x^{2} + 588 x - 2400}}{2 \cdot - 3}$

Step 7.1.10

Subtract $24 x^{2}$ from $x^{2}$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} - 140 x + 4900 + 588 x - 2400}}{2 \cdot - 3}$

Step 7.1.11

Add $- 140 x$ and $588 x$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 4900 - 2400}}{2 \cdot - 3}$

Step 7.1.12

Subtract $2400$ from $4900$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{2 \cdot - 3}$

Step 7.2

Multiply $2$ by $- 3$ .

$y = \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{- 6}$

Step 7.3

Move the negative in front of the fraction.

$y = - \frac{x - 70 \pm \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

Step 7.4

Change the $\pm$ to $-$ .

$y = - \frac{x - 70 - \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

Step 8

The final answer is the combination of both solutions.

$y = - \frac{x - 70 + \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

$y = - \frac{x - 70 - \sqrt{- 23 x^{2} + 448 x + 2500}}{6}$

Step 9

Set the radicand in $\sqrt{- 23 x^{2} + 448 x + 2500}$ greater than or equal to $0$ to find where the expression is defined.

$- 23 x^{2} + 448 x + 2500 \geq 0$

Step 10

Solve for $x$ .

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Step 10.1

Convert the inequality to an equation.

$- 23 x^{2} + 448 x + 2500 = 0$

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 $a = - 23$ , $b = 448$ , and $c = 2500$ into the quadratic formula and solve for $x$ .

$\frac{- 448 \pm \sqrt{448^{2} - 4 \cdot (- 23 \cdot 2500)}}{2 \cdot - 23}$

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 $448$ to the power of $2$ .

$x = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 23 \cdot 2500}}{2 \cdot - 23}$

Step 10.4.1.2

Multiply $- 4 \cdot - 23 \cdot 2500$ .

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Step 10.4.1.2.1

Multiply $- 4$ by $- 23$ .

$x = \frac{- 448 \pm \sqrt{200704 + 92 \cdot 2500}}{2 \cdot - 23}$

Step 10.4.1.2.2

Multiply $92$ by $2500$ .

$x = \frac{- 448 \pm \sqrt{200704 + 230000}}{2 \cdot - 23}$

Step 10.4.1.3

Add $200704$ and $230000$ .

$x = \frac{- 448 \pm \sqrt{430704}}{2 \cdot - 23}$

Step 10.4.1.4

Rewrite $430704$ as $12^{2} \cdot 2991$ .

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Step 10.4.1.4.1

Factor $144$ out of $430704$ .

$x = \frac{- 448 \pm \sqrt{144 (2991)}}{2 \cdot - 23}$

Step 10.4.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 448 \pm \sqrt{12^{2} \cdot 2991}}{2 \cdot - 23}$

Step 10.4.1.5

Pull terms out from under the radical.

$x = \frac{- 448 \pm 12 \sqrt{2991}}{2 \cdot - 23}$

Step 10.4.2

Multiply $2$ by $- 23$ .

$x = \frac{- 448 \pm 12 \sqrt{2991}}{- 46}$

Step 10.4.3

Simplify $\frac{- 448 \pm 12 \sqrt{2991}}{- 46}$ .

$x = \frac{224 \pm 6 \sqrt{2991}}{23}$

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 $448$ to the power of $2$ .

$x = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 23 \cdot 2500}}{2 \cdot - 23}$

Step 10.5.1.2

Multiply $- 4 \cdot - 23 \cdot 2500$ .

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Step 10.5.1.2.1

Multiply $- 4$ by $- 23$ .

$x = \frac{- 448 \pm \sqrt{200704 + 92 \cdot 2500}}{2 \cdot - 23}$

Step 10.5.1.2.2

Multiply $92$ by $2500$ .

$x = \frac{- 448 \pm \sqrt{200704 + 230000}}{2 \cdot - 23}$

Step 10.5.1.3

Add $200704$ and $230000$ .

$x = \frac{- 448 \pm \sqrt{430704}}{2 \cdot - 23}$

Step 10.5.1.4

Rewrite $430704$ as $12^{2} \cdot 2991$ .

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Step 10.5.1.4.1

Factor $144$ out of $430704$ .

$x = \frac{- 448 \pm \sqrt{144 (2991)}}{2 \cdot - 23}$

Step 10.5.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 448 \pm \sqrt{12^{2} \cdot 2991}}{2 \cdot - 23}$

Step 10.5.1.5

Pull terms out from under the radical.

$x = \frac{- 448 \pm 12 \sqrt{2991}}{2 \cdot - 23}$

Step 10.5.2

Multiply $2$ by $- 23$ .

$x = \frac{- 448 \pm 12 \sqrt{2991}}{- 46}$

Step 10.5.3

Simplify $\frac{- 448 \pm 12 \sqrt{2991}}{- 46}$ .

$x = \frac{224 \pm 6 \sqrt{2991}}{23}$

Step 10.5.4

Change the $\pm$ to $+$ .

$x = \frac{224 + 6 \sqrt{2991}}{23}$

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 $448$ to the power of $2$ .

$x = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 23 \cdot 2500}}{2 \cdot - 23}$

Step 10.6.1.2

Multiply $- 4 \cdot - 23 \cdot 2500$ .

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Step 10.6.1.2.1

Multiply $- 4$ by $- 23$ .

$x = \frac{- 448 \pm \sqrt{200704 + 92 \cdot 2500}}{2 \cdot - 23}$

Step 10.6.1.2.2

Multiply $92$ by $2500$ .

$x = \frac{- 448 \pm \sqrt{200704 + 230000}}{2 \cdot - 23}$

Step 10.6.1.3

Add $200704$ and $230000$ .

$x = \frac{- 448 \pm \sqrt{430704}}{2 \cdot - 23}$

Step 10.6.1.4

Rewrite $430704$ as $12^{2} \cdot 2991$ .

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Step 10.6.1.4.1

Factor $144$ out of $430704$ .

$x = \frac{- 448 \pm \sqrt{144 (2991)}}{2 \cdot - 23}$

Step 10.6.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 448 \pm \sqrt{12^{2} \cdot 2991}}{2 \cdot - 23}$

Step 10.6.1.5

Pull terms out from under the radical.

$x = \frac{- 448 \pm 12 \sqrt{2991}}{2 \cdot - 23}$

Step 10.6.2

Multiply $2$ by $- 23$ .

$x = \frac{- 448 \pm 12 \sqrt{2991}}{- 46}$

Step 10.6.3

Simplify $\frac{- 448 \pm 12 \sqrt{2991}}{- 46}$ .

$x = \frac{224 \pm 6 \sqrt{2991}}{23}$

Step 10.6.4

Change the $\pm$ to $-$ .

$x = \frac{224 - 6 \sqrt{2991}}{23}$

Step 10.7

Consolidate the solutions.

$x = \frac{224 + 6 \sqrt{2991}}{23}, \frac{224 - 6 \sqrt{2991}}{23}$

Step 10.8

Use each root to create test intervals.

$x < \frac{224 - 6 \sqrt{2991}}{23}$

$\frac{224 - 6 \sqrt{2991}}{23} < x < \frac{224 + 6 \sqrt{2991}}{23}$

$x > \frac{224 + 6 \sqrt{2991}}{23}$

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 $x < \frac{224 - 6 \sqrt{2991}}{23}$ to see if it makes the inequality true.

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Step 10.9.1.1

Choose a value on the interval $x < \frac{224 - 6 \sqrt{2991}}{23}$ and see if this value makes the original inequality true.

$x = - 7$

Step 10.9.1.2

Replace $x$ with $- 7$ in the original inequality.

$- 23 {(- 7)}^{2} + 448 (- 7) + 2500 \geq 0$

Step 10.9.1.3

The left side $- 1763$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.9.2

Test a value on the interval $\frac{224 - 6 \sqrt{2991}}{23} < x < \frac{224 + 6 \sqrt{2991}}{23}$ to see if it makes the inequality true.

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Step 10.9.2.1

Choose a value on the interval $\frac{224 - 6 \sqrt{2991}}{23} < x < \frac{224 + 6 \sqrt{2991}}{23}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.9.2.2

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

$- 23 {(0)}^{2} + 448 (0) + 2500 \geq 0$

Step 10.9.2.3

The left side $2500$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.9.3

Test a value on the interval $x > \frac{224 + 6 \sqrt{2991}}{23}$ to see if it makes the inequality true.

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Step 10.9.3.1

Choose a value on the interval $x > \frac{224 + 6 \sqrt{2991}}{23}$ and see if this value makes the original inequality true.

$x = 27$

Step 10.9.3.2

Replace $x$ with $27$ in the original inequality.

$- 23 {(27)}^{2} + 448 (27) + 2500 \geq 0$

Step 10.9.3.3

The left side $- 2171$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.9.4

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

$x < \frac{224 - 6 \sqrt{2991}}{23}$ False

$\frac{224 - 6 \sqrt{2991}}{23} < x < \frac{224 + 6 \sqrt{2991}}{23}$ True

$x > \frac{224 + 6 \sqrt{2991}}{23}$ False

$x < \frac{224 - 6 \sqrt{2991}}{23}$ False

$\frac{224 - 6 \sqrt{2991}}{23} < x < \frac{224 + 6 \sqrt{2991}}{23}$ True

$x > \frac{224 + 6 \sqrt{2991}}{23}$ False

Step 10.10

The solution consists of all of the true intervals.

$\frac{224 - 6 \sqrt{2991}}{23} \leq x \leq \frac{224 + 6 \sqrt{2991}}{23}$

Step 11

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

Interval Notation:

$[\frac{224 - 6 \sqrt{2991}}{23}, \frac{224 + 6 \sqrt{2991}}{23}]$

Set-Builder Notation:

${x | \frac{224 - 6 \sqrt{2991}}{23} \leq x \leq \frac{224 + 6 \sqrt{2991}}{23}}$

Step 12

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

Interval Notation:

$[\frac{231 - 6 \sqrt{1994}}{23}, \frac{231 + 6 \sqrt{1994}}{23}]$

Set-Builder Notation:

${y | \frac{231 - 6 \sqrt{1994}}{23} \leq y \leq \frac{231 + 6 \sqrt{1994}}{23}}$

Step 13

Determine the domain and range.

Domain: $[\frac{224 - 6 \sqrt{2991}}{23}, \frac{224 + 6 \sqrt{2991}}{23}], {x | \frac{224 - 6 \sqrt{2991}}{23} \leq x \leq \frac{224 + 6 \sqrt{2991}}{23}}$

Range: $[\frac{231 - 6 \sqrt{1994}}{23}, \frac{231 + 6 \sqrt{1994}}{23}], {y | \frac{231 - 6 \sqrt{1994}}{23} \leq y \leq \frac{231 + 6 \sqrt{1994}}{23}}$

Step 14