Calculus Examples

$- 4 x^{2} - y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 - 3 y^{2} + 5 x^{2} - y^{2} = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1

Simplify $- 4 x^{2} - y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 - 3 y^{2} + 5 x^{2} - y^{2}$ .

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Step 1.1

Add $- 4 x^{2}$ and $5 x^{2}$ .

$x^{2} - y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 - 3 y^{2} - y^{2} = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.2

Subtract $3 y^{2}$ from $- y^{2}$ .

$x^{2} - 4 y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 - y^{2} = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.3

Subtract $y^{2}$ from $- 4 y^{2}$ .

$x^{2} - 5 y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.4

Subtract $10 x$ from $6 x$ .

$x^{2} - 5 y^{2} - 4 x + 2 y - y + 16 - 27 + 3 y + 5 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.5

Subtract $y$ from $2 y$ .

$x^{2} - 5 y^{2} - 4 x + y + 16 - 27 + 3 y + 5 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.6

Add $y$ and $3 y$ .

$x^{2} - 5 y^{2} - 4 x + 4 y + 16 - 27 + 5 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.7

Simplify by adding and subtracting.

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Step 1.7.1

Subtract $27$ from $16$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 11 + 5 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1.7.2

Add $- 11$ and $5$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 2

Simplify $9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$ .

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Step 2.1

Subtract $30$ from $9$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = - 6 y^{2} - 4 y - 21 + 10 x + 17 + y - 3 y - 60$

Step 2.2

Add $- 4 y$ and $y$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = - 6 y^{2} - 3 y - 21 + 10 x + 17 - 3 y - 60$

Step 2.3

Subtract $3 y$ from $- 3 y$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = - 6 y^{2} - 6 y - 21 + 10 x + 17 - 60$

Step 2.4

Simplify by adding and subtracting.

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Step 2.4.1

Add $- 21$ and $17$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = - 6 y^{2} - 6 y + 10 x - 4 - 60$

Step 2.4.2

Subtract $60$ from $- 4$ .

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 = - 6 y^{2} - 6 y + 10 x - 64$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $10 x$ from both sides of the equation.

$x^{2} - 5 y^{2} - 4 x + 4 y - 6 - 10 x = - 6 y^{2} - 6 y - 64$

Step 3.2

Subtract $10 x$ from $- 4 x$ .

$x^{2} - 5 y^{2} - 14 x + 4 y - 6 = - 6 y^{2} - 6 y - 64$

Step 4

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

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Step 4.1

Move all the expressions to the left side of the equation.

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Step 4.1.1

Add $6 y^{2}$ to both sides of the equation.

$x^{2} - 5 y^{2} - 14 x + 4 y - 6 + 6 y^{2} = - 6 y - 64$

Step 4.1.2

Add $6 y$ to both sides of the equation.

$x^{2} - 5 y^{2} - 14 x + 4 y - 6 + 6 y^{2} + 6 y = - 64$

Step 4.1.3

Add $64$ to both sides of the equation.

$x^{2} - 5 y^{2} - 14 x + 4 y - 6 + 6 y^{2} + 6 y + 64 = 0$

Step 4.2

Simplify $x^{2} - 5 y^{2} - 14 x + 4 y - 6 + 6 y^{2} + 6 y + 64$ .

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Step 4.2.1

Add $- 5 y^{2}$ and $6 y^{2}$ .

$x^{2} + y^{2} - 14 x + 4 y - 6 + 6 y + 64 = 0$

Step 4.2.2

Add $4 y$ and $6 y$ .

$x^{2} + y^{2} - 14 x + 10 y - 6 + 64 = 0$

Step 4.2.3

Add $- 6$ and $64$ .

$x^{2} + y^{2} - 14 x + 10 y + 58 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 14$ , and $c = y^{2} + 10 y + 58$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot (y^{2} + 10 y + 58))}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot (y^{2} + 10 y + 58)}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot (y^{2} + 10 y + 58)}}{2 \cdot 1}$

Step 7.1.3

Apply the distributive property.

$x = \frac{14 \pm \sqrt{196 - 4 y^{2} - 4 (10 y) - 4 \cdot 58}}{2 \cdot 1}$

Step 7.1.4

Simplify.

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Step 7.1.4.1

Multiply $10$ by $- 4$ .

$x = \frac{14 \pm \sqrt{196 - 4 y^{2} - 40 y - 4 \cdot 58}}{2 \cdot 1}$

Step 7.1.4.2

Multiply $- 4$ by $58$ .

$x = \frac{14 \pm \sqrt{196 - 4 y^{2} - 40 y - 232}}{2 \cdot 1}$

Step 7.1.5

Subtract $232$ from $196$ .

$x = \frac{14 \pm \sqrt{- 4 y^{2} - 40 y - 36}}{2 \cdot 1}$

Step 7.1.6

Rewrite $- 4 y^{2} - 40 y - 36$ in a factored form.

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Step 7.1.6.1

Factor $4$ out of $- 4 y^{2} - 40 y - 36$ .

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Step 7.1.6.1.1

Factor $4$ out of $- 4 y^{2}$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2}) - 40 y - 36}}{2 \cdot 1}$

Step 7.1.6.1.2

Factor $4$ out of $- 40 y$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2}) + 4 (- 10 y) - 36}}{2 \cdot 1}$

Step 7.1.6.1.3

Factor $4$ out of $- 36$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2}) + 4 (- 10 y) + 4 (- 9)}}{2 \cdot 1}$

Step 7.1.6.1.4

Factor $4$ out of $4 (- y^{2}) + 4 (- 10 y)$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2} - 10 y) + 4 (- 9)}}{2 \cdot 1}$

Step 7.1.6.1.5

Factor $4$ out of $4 (- y^{2} - 10 y) + 4 (- 9)$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2} - 10 y - 9)}}{2 \cdot 1}$

Step 7.1.6.2

Factor by grouping.

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Step 7.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = - 10$ .

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Step 7.1.6.2.1.1

Factor $- 10$ out of $- 10 y$ .

$x = \frac{14 \pm \sqrt{4 (- y^{2} - 10 y - 9)}}{2 \cdot 1}$

Step 7.1.6.2.1.2

Rewrite $- 10$ as $- 1$ plus $- 9$

$x = \frac{14 \pm \sqrt{4 (- y^{2} + (- 1 - 9) y - 9)}}{2 \cdot 1}$

Step 7.1.6.2.1.3

Apply the distributive property.

$x = \frac{14 \pm \sqrt{4 (- y^{2} - 1 y - 9 y - 9)}}{2 \cdot 1}$

Step 7.1.6.2.2

Factor out the greatest common factor from each group.

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Step 7.1.6.2.2.1

Group the first two terms and the last two terms.

$x = \frac{14 \pm \sqrt{4 ((- y^{2} - 1 y) - 9 y - 9)}}{2 \cdot 1}$

Step 7.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$x = \frac{14 \pm \sqrt{4 (y (- y - 1) + 9 (- y - 1))}}{2 \cdot 1}$

Step 7.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- y - 1$ .

$x = \frac{14 \pm \sqrt{4 ((- y - 1) (y + 9))}}{2 \cdot 1}$

$x = \frac{14 \pm \sqrt{4 (- y - 1) (y + 9)}}{2 \cdot 1}$

Step 7.1.7

Rewrite $4 (- y - 1) (y + 9)$ as $2^{2} ((- y - 1) (y + 3^{2}))$ .

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Step 7.1.7.1

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

$x = \frac{14 \pm \sqrt{2^{2} (- y - 1) (y + 9)}}{2 \cdot 1}$

Step 7.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{14 \pm \sqrt{2^{2} (- y - 1) (y + 3^{2})}}{2 \cdot 1}$

Step 7.1.7.3

Add parentheses.

$x = \frac{14 \pm \sqrt{2^{2} ((- y - 1) (y + 3^{2}))}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{(- y - 1) (y + 3^{2})}}{2 \cdot 1}$

Step 7.1.9

Raise $3$ to the power of $2$ .

$x = \frac{14 \pm 2 \sqrt{(- y - 1) (y + 9)}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{14 \pm 2 \sqrt{(- y - 1) (y + 9)}}{2}$

Step 7.3

Simplify $\frac{14 \pm 2 \sqrt{(- y - 1) (y + 9)}}{2}$ .

$x = 7 \pm \sqrt{(- y - 1) (y + 9)}$

Step 8

The final answer is the combination of both solutions.

$x = 7 + \sqrt{(- y - 1) (y + 9)}$

$x = 7 - \sqrt{(- y - 1) (y + 9)}$