Precalculus Examples

$x^{2} - 9 y^{2} + 2 x - 54 y - 80 = 0$

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 $a = - 9$ , $b = - 54$ , and $c = x^{2} + 2 x - 80$ into the quadratic formula and solve for $y$ .

$\frac{54 \pm \sqrt{{(- 54)}^{2} - 4 \cdot (- 9 \cdot (x^{2} + 2 x - 80))}}{2 \cdot - 9}$

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 $- 54$ to the power of $2$ .

$y = \frac{54 \pm \sqrt{2916 - 4 \cdot - 9 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.3.1.2

Multiply $- 4$ by $- 9$ .

$y = \frac{54 \pm \sqrt{2916 + 36 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 36 (2 x) + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $2$ by $36$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.3.1.4.2

Multiply $36$ by $- 80$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x - 2880}}{2 \cdot - 9}$

Step 1.3.1.5

Subtract $2880$ from $2916$ .

$y = \frac{54 \pm \sqrt{36 x^{2} + 72 x + 36}}{2 \cdot - 9}$

Step 1.3.1.6

Rewrite $36 x^{2} + 72 x + 36$ in a factored form.

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Step 1.3.1.6.1

Factor $36$ out of $36 x^{2} + 72 x + 36$ .

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Step 1.3.1.6.1.1

Factor $36$ out of $36 x^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 72 x + 36}}{2 \cdot - 9}$

Step 1.3.1.6.1.2

Factor $36$ out of $72 x$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36}}{2 \cdot - 9}$

Step 1.3.1.6.1.3

Factor $36$ out of $36$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.3.1.6.1.4

Factor $36$ out of $36 (x^{2}) + 36 (2 x)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.3.1.6.1.5

Factor $36$ out of $36 (x^{2} + 2 x) + 36 (1)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1)}}{2 \cdot - 9}$

Step 1.3.1.6.2

Factor using the perfect square rule.

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Step 1.3.1.6.2.1

Rewrite $1$ as $1^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1^{2})}}{2 \cdot - 9}$

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.

$2 x = 2 \cdot x \cdot 1$

Step 1.3.1.6.2.3

Rewrite the polynomial.

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot - 9}$

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 $b = 1$ .

$y = \frac{54 \pm \sqrt{36 {(x + 1)}^{2}}}{2 \cdot - 9}$

Step 1.3.1.7

Rewrite $36 {(x + 1)}^{2}$ as ${(6 (x + 1))}^{2}$ .

$y = \frac{54 \pm \sqrt{{(6 (x + 1))}^{2}}}{2 \cdot - 9}$

Step 1.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{54 \pm 6 (x + 1)}{2 \cdot - 9}$

Step 1.3.1.9

Apply the distributive property.

$y = \frac{54 \pm (6 x + 6 \cdot 1)}{2 \cdot - 9}$

Step 1.3.1.10

Multiply $6$ by $1$ .

$y = \frac{54 \pm (6 x + 6)}{2 \cdot - 9}$

Step 1.3.2

Multiply $2$ by $- 9$ .

$y = \frac{54 \pm (6 x + 6)}{- 18}$

Step 1.3.3

Move the negative in front of the fraction.

$y = - \frac{54 \pm (6 x + 6)}{18}$

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 $- 54$ to the power of $2$ .

$y = \frac{54 \pm \sqrt{2916 - 4 \cdot - 9 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.4.1.2

Multiply $- 4$ by $- 9$ .

$y = \frac{54 \pm \sqrt{2916 + 36 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 36 (2 x) + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $2$ by $36$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.4.1.4.2

Multiply $36$ by $- 80$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x - 2880}}{2 \cdot - 9}$

Step 1.4.1.5

Subtract $2880$ from $2916$ .

$y = \frac{54 \pm \sqrt{36 x^{2} + 72 x + 36}}{2 \cdot - 9}$

Step 1.4.1.6

Rewrite $36 x^{2} + 72 x + 36$ in a factored form.

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Step 1.4.1.6.1

Factor $36$ out of $36 x^{2} + 72 x + 36$ .

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Step 1.4.1.6.1.1

Factor $36$ out of $36 x^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 72 x + 36}}{2 \cdot - 9}$

Step 1.4.1.6.1.2

Factor $36$ out of $72 x$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36}}{2 \cdot - 9}$

Step 1.4.1.6.1.3

Factor $36$ out of $36$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.4.1.6.1.4

Factor $36$ out of $36 (x^{2}) + 36 (2 x)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.4.1.6.1.5

Factor $36$ out of $36 (x^{2} + 2 x) + 36 (1)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1)}}{2 \cdot - 9}$

Step 1.4.1.6.2

Factor using the perfect square rule.

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Step 1.4.1.6.2.1

Rewrite $1$ as $1^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1^{2})}}{2 \cdot - 9}$

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.

$2 x = 2 \cdot x \cdot 1$

Step 1.4.1.6.2.3

Rewrite the polynomial.

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot - 9}$

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 $b = 1$ .

$y = \frac{54 \pm \sqrt{36 {(x + 1)}^{2}}}{2 \cdot - 9}$

Step 1.4.1.7

Rewrite $36 {(x + 1)}^{2}$ as ${(6 (x + 1))}^{2}$ .

$y = \frac{54 \pm \sqrt{{(6 (x + 1))}^{2}}}{2 \cdot - 9}$

Step 1.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{54 \pm 6 (x + 1)}{2 \cdot - 9}$

Step 1.4.1.9

Apply the distributive property.

$y = \frac{54 \pm (6 x + 6 \cdot 1)}{2 \cdot - 9}$

Step 1.4.1.10

Multiply $6$ by $1$ .

$y = \frac{54 \pm (6 x + 6)}{2 \cdot - 9}$

Step 1.4.2

Multiply $2$ by $- 9$ .

$y = \frac{54 \pm (6 x + 6)}{- 18}$

Step 1.4.3

Move the negative in front of the fraction.

$y = - \frac{54 \pm (6 x + 6)}{18}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = - \frac{54 + 6 x + 6}{18}$

Step 1.4.5

Cancel the common factor of $54 + 6 x + 6$ and $18$ .

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Step 1.4.5.1

Factor $6$ out of $54$ .

$y = - \frac{6 \cdot 9 + 6 x + 6}{18}$

Step 1.4.5.2

Factor $6$ out of $6 x$ .

$y = - \frac{6 \cdot 9 + 6 (x) + 6}{18}$

Step 1.4.5.3

Factor $6$ out of $6$ .

$y = - \frac{6 \cdot 9 + 6 (x) + 6 (1)}{18}$

Step 1.4.5.4

Factor $6$ out of $6 (x) + 6 (1)$ .

$y = - \frac{6 \cdot 9 + 6 (x + 1)}{18}$

Step 1.4.5.5

Factor $6$ out of $6 \cdot 9 + 6 (x + 1)$ .

$y = - \frac{6 \cdot (9 + x + 1)}{18}$

Step 1.4.5.6

Cancel the common factors.

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Step 1.4.5.6.1

Factor $6$ out of $18$ .

$y = - \frac{6 \cdot (9 + x + 1)}{6 (3)}$

Step 1.4.5.6.2

Cancel the common factor.

$y = - \frac{6 \cdot (9 + x + 1)}{6 \cdot 3}$

Step 1.4.5.6.3

Rewrite the expression.

$y = - \frac{9 + x + 1}{3}$

Step 1.4.6

Add $9$ and $1$ .

$y = - \frac{x + 10}{3}$

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 $- 54$ to the power of $2$ .

$y = \frac{54 \pm \sqrt{2916 - 4 \cdot - 9 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.5.1.2

Multiply $- 4$ by $- 9$ .

$y = \frac{54 \pm \sqrt{2916 + 36 \cdot (x^{2} + 2 x - 80)}}{2 \cdot - 9}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 36 (2 x) + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $2$ by $36$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x + 36 \cdot - 80}}{2 \cdot - 9}$

Step 1.5.1.4.2

Multiply $36$ by $- 80$ .

$y = \frac{54 \pm \sqrt{2916 + 36 x^{2} + 72 x - 2880}}{2 \cdot - 9}$

Step 1.5.1.5

Subtract $2880$ from $2916$ .

$y = \frac{54 \pm \sqrt{36 x^{2} + 72 x + 36}}{2 \cdot - 9}$

Step 1.5.1.6

Rewrite $36 x^{2} + 72 x + 36$ in a factored form.

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Step 1.5.1.6.1

Factor $36$ out of $36 x^{2} + 72 x + 36$ .

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Step 1.5.1.6.1.1

Factor $36$ out of $36 x^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 72 x + 36}}{2 \cdot - 9}$

Step 1.5.1.6.1.2

Factor $36$ out of $72 x$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36}}{2 \cdot - 9}$

Step 1.5.1.6.1.3

Factor $36$ out of $36$ .

$y = \frac{54 \pm \sqrt{36 (x^{2}) + 36 (2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.5.1.6.1.4

Factor $36$ out of $36 (x^{2}) + 36 (2 x)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x) + 36 (1)}}{2 \cdot - 9}$

Step 1.5.1.6.1.5

Factor $36$ out of $36 (x^{2} + 2 x) + 36 (1)$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1)}}{2 \cdot - 9}$

Step 1.5.1.6.2

Factor using the perfect square rule.

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Step 1.5.1.6.2.1

Rewrite $1$ as $1^{2}$ .

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 x + 1^{2})}}{2 \cdot - 9}$

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.

$2 x = 2 \cdot x \cdot 1$

Step 1.5.1.6.2.3

Rewrite the polynomial.

$y = \frac{54 \pm \sqrt{36 (x^{2} + 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot - 9}$

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 $b = 1$ .

$y = \frac{54 \pm \sqrt{36 {(x + 1)}^{2}}}{2 \cdot - 9}$

Step 1.5.1.7

Rewrite $36 {(x + 1)}^{2}$ as ${(6 (x + 1))}^{2}$ .

$y = \frac{54 \pm \sqrt{{(6 (x + 1))}^{2}}}{2 \cdot - 9}$

Step 1.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{54 \pm 6 (x + 1)}{2 \cdot - 9}$

Step 1.5.1.9

Apply the distributive property.

$y = \frac{54 \pm (6 x + 6 \cdot 1)}{2 \cdot - 9}$

Step 1.5.1.10

Multiply $6$ by $1$ .

$y = \frac{54 \pm (6 x + 6)}{2 \cdot - 9}$

Step 1.5.2

Multiply $2$ by $- 9$ .

$y = \frac{54 \pm (6 x + 6)}{- 18}$

Step 1.5.3

Move the negative in front of the fraction.

$y = - \frac{54 \pm (6 x + 6)}{18}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = - \frac{54 - (6 x + 6)}{18}$

Step 1.5.5

Cancel the common factor of $54 - (6 x + 6)$ and $18$ .

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Step 1.5.5.1

Rewrite $54$ as $- 1 (- 54)$ .

$y = - \frac{- 1 \cdot - 54 - (6 x + 6)}{18}$

Step 1.5.5.2

Factor $- 1$ out of $- 1 (- 54) - (6 x + 6)$ .

$y = - \frac{- 1 (- 54 + 6 x + 6)}{18}$

Step 1.5.5.3

Factor $6$ out of $- 1 (- 54 + 6 x + 6)$ .

$y = - \frac{6 (- 1 (- 9 + x + 1))}{18}$

Step 1.5.5.4

Cancel the common factors.

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Step 1.5.5.4.1

Factor $6$ out of $18$ .

$y = - \frac{6 (- 1 (- 9 + x + 1))}{6 (3)}$

Step 1.5.5.4.2

Cancel the common factor.

$y = - \frac{6 (- 1 (- 9 + x + 1))}{6 \cdot 3}$

Step 1.5.5.4.3

Rewrite the expression.

$y = - \frac{- 1 (- 9 + x + 1)}{3}$

Step 1.5.6

Add $- 9$ and $1$ .

$y = - \frac{- 1 (x - 8)}{3}$

Step 1.5.7

Move the negative in front of the fraction.

$y = \frac{x - 8}{3}$

Step 1.5.8

Multiply $- - \frac{x - 8}{3}$ .

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Step 1.5.8.1

Multiply $- 1$ by $- 1$ .

$y = 1 (\frac{x - 8}{3})$

Step 1.5.8.2

Multiply $\frac{x - 8}{3}$ by $1$ .

$y = \frac{x - 8}{3}$

Step 1.6

The final answer is the combination of both solutions.

$y = - \frac{x + 10}{3}$

$y = \frac{x - 8}{3}$

$y = - \frac{x + 10}{3}$

$y = \frac{x - 8}{3}$

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 $\frac{x + 10}{3}$ into two fractions.

$y = - (\frac{x}{3} + \frac{10}{3})$

$y = \frac{x - 8}{3}$

Step 4

Apply the distributive property.

$y = - \frac{x}{3} - \frac{10}{3}$

$y = \frac{x - 8}{3}$

Step 5

Split the fraction $\frac{x - 8}{3}$ into two fractions.

$y = - \frac{x}{3} - \frac{10}{3}$

$y = \frac{x}{3} + \frac{- 8}{3}$

Step 6

Move the negative in front of the fraction.

$y = - \frac{x}{3} - \frac{10}{3}$

$y = \frac{x}{3} - \frac{8}{3}$

Step 7

Reorder terms.

$y = - (\frac{1}{3} x) - \frac{10}{3}$

$y = \frac{1}{3} x - \frac{8}{3}$

Step 8

Remove parentheses.

$y = - \frac{1}{3} x - \frac{10}{3}$

$y = \frac{1}{3} x - \frac{8}{3}$

Step 9