Precalculus Examples

$16 y^{2} - x^{2} + 2 x + 64 y + 63 = 0$

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 $a = 16$ , $b = 64$ , and $c = - x^{2} + 2 x + 63$ into the quadratic formula and solve for $y$ .

$\frac{- 64 \pm \sqrt{64^{2} - 4 \cdot (16 \cdot (- x^{2} + 2 x + 63))}}{2 \cdot 16}$

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 $64$ to the power of $2$ .

$y = \frac{- 64 \pm \sqrt{4096 - 4 \cdot 16 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.3.1.2

Multiply $- 4$ by $16$ .

$y = \frac{- 64 \pm \sqrt{4096 - 64 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 64 \pm \sqrt{4096 - 64 (- x^{2}) - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $- 1$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.3.1.4.2

Multiply $2$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 64 \cdot 63}}{2 \cdot 16}$

Step 1.3.1.4.3

Multiply $- 64$ by $63$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 4032}}{2 \cdot 16}$

Step 1.3.1.5

Subtract $4032$ from $4096$ .

$y = \frac{- 64 \pm \sqrt{64 x^{2} - 128 x + 64}}{2 \cdot 16}$

Step 1.3.1.6

Rewrite $64 x^{2} - 128 x + 64$ in a factored form.

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Step 1.3.1.6.1

Factor $64$ out of $64 x^{2} - 128 x + 64$ .

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Step 1.3.1.6.1.1

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

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) - 128 x + 64}}{2 \cdot 16}$

Step 1.3.1.6.1.2

Factor $64$ out of $- 128 x$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64}}{2 \cdot 16}$

Step 1.3.1.6.1.3

Factor $64$ out of $64$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.3.1.6.1.4

Factor $64$ out of $64 (x^{2}) + 64 (- 2 x)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.3.1.6.1.5

Factor $64$ out of $64 (x^{2} - 2 x) + 64 (1)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1)}}{2 \cdot 16}$

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 $1^{2}$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1^{2})}}{2 \cdot 16}$

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.

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot 16}$

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 $b = 1$ .

$y = \frac{- 64 \pm \sqrt{64 {(x - 1)}^{2}}}{2 \cdot 16}$

Step 1.3.1.7

Rewrite $64 {(x - 1)}^{2}$ as ${(8 (x - 1))}^{2}$ .

$y = \frac{- 64 \pm \sqrt{{(8 (x - 1))}^{2}}}{2 \cdot 16}$

Step 1.3.1.8

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

$y = \frac{- 64 \pm 8 (x - 1)}{2 \cdot 16}$

Step 1.3.1.9

Apply the distributive property.

$y = \frac{- 64 \pm (8 x + 8 \cdot - 1)}{2 \cdot 16}$

Step 1.3.1.10

Multiply $8$ by $- 1$ .

$y = \frac{- 64 \pm (8 x - 8)}{2 \cdot 16}$

Step 1.3.2

Multiply $2$ by $16$ .

$y = \frac{- 64 \pm (8 x - 8)}{32}$

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 $64$ to the power of $2$ .

$y = \frac{- 64 \pm \sqrt{4096 - 4 \cdot 16 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.4.1.2

Multiply $- 4$ by $16$ .

$y = \frac{- 64 \pm \sqrt{4096 - 64 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 64 \pm \sqrt{4096 - 64 (- x^{2}) - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $- 1$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.4.1.4.2

Multiply $2$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 64 \cdot 63}}{2 \cdot 16}$

Step 1.4.1.4.3

Multiply $- 64$ by $63$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 4032}}{2 \cdot 16}$

Step 1.4.1.5

Subtract $4032$ from $4096$ .

$y = \frac{- 64 \pm \sqrt{64 x^{2} - 128 x + 64}}{2 \cdot 16}$

Step 1.4.1.6

Rewrite $64 x^{2} - 128 x + 64$ in a factored form.

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Step 1.4.1.6.1

Factor $64$ out of $64 x^{2} - 128 x + 64$ .

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Step 1.4.1.6.1.1

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

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) - 128 x + 64}}{2 \cdot 16}$

Step 1.4.1.6.1.2

Factor $64$ out of $- 128 x$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64}}{2 \cdot 16}$

Step 1.4.1.6.1.3

Factor $64$ out of $64$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.4.1.6.1.4

Factor $64$ out of $64 (x^{2}) + 64 (- 2 x)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.4.1.6.1.5

Factor $64$ out of $64 (x^{2} - 2 x) + 64 (1)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1)}}{2 \cdot 16}$

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 $1^{2}$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1^{2})}}{2 \cdot 16}$

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.

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot 16}$

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 $b = 1$ .

$y = \frac{- 64 \pm \sqrt{64 {(x - 1)}^{2}}}{2 \cdot 16}$

Step 1.4.1.7

Rewrite $64 {(x - 1)}^{2}$ as ${(8 (x - 1))}^{2}$ .

$y = \frac{- 64 \pm \sqrt{{(8 (x - 1))}^{2}}}{2 \cdot 16}$

Step 1.4.1.8

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

$y = \frac{- 64 \pm 8 (x - 1)}{2 \cdot 16}$

Step 1.4.1.9

Apply the distributive property.

$y = \frac{- 64 \pm (8 x + 8 \cdot - 1)}{2 \cdot 16}$

Step 1.4.1.10

Multiply $8$ by $- 1$ .

$y = \frac{- 64 \pm (8 x - 8)}{2 \cdot 16}$

Step 1.4.2

Multiply $2$ by $16$ .

$y = \frac{- 64 \pm (8 x - 8)}{32}$

Step 1.4.3

Change the $\pm$ to $+$ .

$y = \frac{- 64 + 8 x - 8}{32}$

Step 1.4.4

Cancel the common factor of $- 64 + 8 x - 8$ and $32$ .

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Step 1.4.4.1

Factor $8$ out of $- 64$ .

$y = \frac{8 (- 8) + 8 x - 8}{32}$

Step 1.4.4.2

Factor $8$ out of $8 x$ .

$y = \frac{8 (- 8) + 8 (x) - 8}{32}$

Step 1.4.4.3

Factor $8$ out of $- 8$ .

$y = \frac{8 \cdot - 8 + 8 (x) + 8 \cdot - 1}{32}$

Step 1.4.4.4

Factor $8$ out of $8 (x) + 8 (- 1)$ .

$y = \frac{8 (- 8) + 8 (x - 1)}{32}$

Step 1.4.4.5

Factor $8$ out of $8 (- 8) + 8 (x - 1)$ .

$y = \frac{8 (- 8 + x - 1)}{32}$

Step 1.4.4.6

Cancel the common factors.

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Step 1.4.4.6.1

Factor $8$ out of $32$ .

$y = \frac{8 (- 8 + x - 1)}{8 \cdot 4}$

Step 1.4.4.6.2

Cancel the common factor.

$y = \frac{8 (- 8 + x - 1)}{8 \cdot 4}$

Step 1.4.4.6.3

Rewrite the expression.

$y = \frac{- 8 + x - 1}{4}$

Step 1.4.5

Subtract $1$ from $- 8$ .

$y = \frac{x - 9}{4}$

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 $64$ to the power of $2$ .

$y = \frac{- 64 \pm \sqrt{4096 - 4 \cdot 16 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.5.1.2

Multiply $- 4$ by $16$ .

$y = \frac{- 64 \pm \sqrt{4096 - 64 \cdot (- x^{2} + 2 x + 63)}}{2 \cdot 16}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 64 \pm \sqrt{4096 - 64 (- x^{2}) - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $- 1$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 64 (2 x) - 64 \cdot 63}}{2 \cdot 16}$

Step 1.5.1.4.2

Multiply $2$ by $- 64$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 64 \cdot 63}}{2 \cdot 16}$

Step 1.5.1.4.3

Multiply $- 64$ by $63$ .

$y = \frac{- 64 \pm \sqrt{4096 + 64 x^{2} - 128 x - 4032}}{2 \cdot 16}$

Step 1.5.1.5

Subtract $4032$ from $4096$ .

$y = \frac{- 64 \pm \sqrt{64 x^{2} - 128 x + 64}}{2 \cdot 16}$

Step 1.5.1.6

Rewrite $64 x^{2} - 128 x + 64$ in a factored form.

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Step 1.5.1.6.1

Factor $64$ out of $64 x^{2} - 128 x + 64$ .

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Step 1.5.1.6.1.1

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

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) - 128 x + 64}}{2 \cdot 16}$

Step 1.5.1.6.1.2

Factor $64$ out of $- 128 x$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64}}{2 \cdot 16}$

Step 1.5.1.6.1.3

Factor $64$ out of $64$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2}) + 64 (- 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.5.1.6.1.4

Factor $64$ out of $64 (x^{2}) + 64 (- 2 x)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x) + 64 (1)}}{2 \cdot 16}$

Step 1.5.1.6.1.5

Factor $64$ out of $64 (x^{2} - 2 x) + 64 (1)$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1)}}{2 \cdot 16}$

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 $1^{2}$ .

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 x + 1^{2})}}{2 \cdot 16}$

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.

$y = \frac{- 64 \pm \sqrt{64 (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}}{2 \cdot 16}$

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 $b = 1$ .

$y = \frac{- 64 \pm \sqrt{64 {(x - 1)}^{2}}}{2 \cdot 16}$

Step 1.5.1.7

Rewrite $64 {(x - 1)}^{2}$ as ${(8 (x - 1))}^{2}$ .

$y = \frac{- 64 \pm \sqrt{{(8 (x - 1))}^{2}}}{2 \cdot 16}$

Step 1.5.1.8

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

$y = \frac{- 64 \pm 8 (x - 1)}{2 \cdot 16}$

Step 1.5.1.9

Apply the distributive property.

$y = \frac{- 64 \pm (8 x + 8 \cdot - 1)}{2 \cdot 16}$

Step 1.5.1.10

Multiply $8$ by $- 1$ .

$y = \frac{- 64 \pm (8 x - 8)}{2 \cdot 16}$

Step 1.5.2

Multiply $2$ by $16$ .

$y = \frac{- 64 \pm (8 x - 8)}{32}$

Step 1.5.3

Change the $\pm$ to $-$ .

$y = \frac{- 64 - (8 x - 8)}{32}$

Step 1.5.4

Cancel the common factor of $- 64 - (8 x - 8)$ and $32$ .

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Step 1.5.4.1

Rewrite $- 64$ as $- 1 (64)$ .

$y = \frac{- 1 \cdot 64 - (8 x - 8)}{32}$

Step 1.5.4.2

Factor $- 1$ out of $- 1 (64) - (8 x - 8)$ .

$y = \frac{- 1 (64 + 8 x - 8)}{32}$

Step 1.5.4.3

Factor $8$ out of $- 1 (64 + 8 x - 8)$ .

$y = \frac{8 (- 1 (8 + x - 1))}{32}$

Step 1.5.4.4

Cancel the common factors.

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Step 1.5.4.4.1

Factor $8$ out of $32$ .

$y = \frac{8 (- 1 (8 + x - 1))}{8 (4)}$

Step 1.5.4.4.2

Cancel the common factor.

$y = \frac{8 (- 1 (8 + x - 1))}{8 \cdot 4}$

Step 1.5.4.4.3

Rewrite the expression.

$y = \frac{- 1 (8 + x - 1)}{4}$

Step 1.5.5

Subtract $1$ from $8$ .

$y = \frac{- 1 (x + 7)}{4}$

Step 1.5.6

Move the negative in front of the fraction.

$y = - \frac{x + 7}{4}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{x - 9}{4}$

$y = - \frac{x + 7}{4}$

$y = \frac{x - 9}{4}$

$y = - \frac{x + 7}{4}$

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 $\frac{x - 9}{4}$ into two fractions.

$y = \frac{x}{4} + \frac{- 9}{4}$

$y = - \frac{x + 7}{4}$

Step 4

Move the negative in front of the fraction.

$y = \frac{x}{4} - \frac{9}{4}$

$y = - \frac{x + 7}{4}$

Step 5

Split the fraction $\frac{x + 7}{4}$ into two fractions.

$y = \frac{x}{4} - \frac{9}{4}$

$y = - (\frac{x}{4} + \frac{7}{4})$

Step 6

Apply the distributive property.

$y = \frac{x}{4} - \frac{9}{4}$

$y = - \frac{x}{4} - \frac{7}{4}$

Step 7

Reorder terms.

$y = \frac{1}{4} x - \frac{9}{4}$

$y = - (\frac{1}{4} x) - \frac{7}{4}$

Step 8

Remove parentheses.

$y = \frac{1}{4} x - \frac{9}{4}$

$y = - \frac{1}{4} x - \frac{7}{4}$

Step 9