Precalculus Examples

$5 x^{2} - 6 x y + 5 y^{2} - 12 = 0$

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 $a = 5$ , $b = - 6 x$ , and $c = 5 x^{2} - 12$ into the quadratic formula and solve for $y$ .

$\frac{6 x \pm \sqrt{{(- 6 x)}^{2} - 4 \cdot (5 \cdot (5 x^{2} - 12))}}{2 \cdot 5}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Add parentheses.

$y = \frac{6 x \pm \sqrt{{(- 6 x)}^{2} - 4 \cdot (5 \cdot (5 x^{2} - 12))}}{2 \cdot 5}$

Step 1.3.1.2

Let $u = 5 \cdot (5 x^{2} - 12)$ . Substitute $u$ for all occurrences of $5 \cdot (5 x^{2} - 12)$ .

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Step 1.3.1.2.1

Apply the product rule to $- 6 x$ .

$y = \frac{6 x \pm \sqrt{{(- 6)}^{2} x^{2} - 4 \cdot u}}{2 \cdot 5}$

Step 1.3.1.2.2

Raise $- 6$ to the power of $2$ .

$y = \frac{6 x \pm \sqrt{36 x^{2} - 4 u}}{2 \cdot 5}$

Step 1.3.1.3

Factor $4$ out of $36 x^{2} - 4 u$ .

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Step 1.3.1.3.1

Factor $4$ out of $36 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) - 4 u}}{2 \cdot 5}$

Step 1.3.1.3.2

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

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) + 4 (- u)}}{2 \cdot 5}$

Step 1.3.1.3.3

Factor $4$ out of $4 (9 x^{2}) + 4 (- u)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - u)}}{2 \cdot 5}$

Step 1.3.1.4

Replace all occurrences of $u$ with $5 \cdot (5 x^{2} - 12)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 \cdot (5 x^{2} - 12)))}}{2 \cdot 5}$

Step 1.3.1.5

Simplify.

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Step 1.3.1.5.1

Simplify each term.

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Step 1.3.1.5.1.1

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 (5 x^{2}) + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.3.1.5.1.2

Multiply $5$ by $5$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.3.1.5.1.3

Multiply $5$ by $- 12$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} - 60))}}{2 \cdot 5}$

Step 1.3.1.5.1.4

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.3.1.5.1.5

Multiply $25$ by $- 1$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.3.1.5.1.6

Multiply $- 1$ by $- 60$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.3.1.5.2

Subtract $25 x^{2}$ from $9 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (- 16 x^{2} + 60)}}{2 \cdot 5}$

Step 1.3.1.6

Factor $4$ out of $- 16 x^{2} + 60$ .

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Step 1.3.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.3.1.6.2

Factor $4$ out of $60$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 4 (15))}}{2 \cdot 5}$

Step 1.3.1.6.3

Factor $4$ out of $4 (- 4 x^{2}) + 4 (15)$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

$y = \frac{6 x \pm \sqrt{4 \cdot (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

Step 1.3.1.7

Multiply $4$ by $4$ .

$y = \frac{6 x \pm \sqrt{16 (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.3.1.8

Rewrite $16 (- 4 x^{2} + 15)$ as ${(2^{2})}^{2} (- 4 x^{2} + 15)$ .

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Step 1.3.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{6 x \pm \sqrt{4^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.3.1.8.2

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

$y = \frac{6 x \pm \sqrt{{(2^{2})}^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{6 x \pm 2^{2} \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.3.1.10

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

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.3.2

Multiply $2$ by $5$ .

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$

Step 1.3.3

Simplify $\frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$ .

$y = \frac{3 x \pm 2 \sqrt{- 4 x^{2} + 15}}{5}$

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

Add parentheses.

$y = \frac{6 x \pm \sqrt{{(- 6 x)}^{2} - 4 \cdot (5 \cdot (5 x^{2} - 12))}}{2 \cdot 5}$

Step 1.4.1.2

Let $u = 5 \cdot (5 x^{2} - 12)$ . Substitute $u$ for all occurrences of $5 \cdot (5 x^{2} - 12)$ .

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Step 1.4.1.2.1

Apply the product rule to $- 6 x$ .

$y = \frac{6 x \pm \sqrt{{(- 6)}^{2} x^{2} - 4 \cdot u}}{2 \cdot 5}$

Step 1.4.1.2.2

Raise $- 6$ to the power of $2$ .

$y = \frac{6 x \pm \sqrt{36 x^{2} - 4 u}}{2 \cdot 5}$

Step 1.4.1.3

Factor $4$ out of $36 x^{2} - 4 u$ .

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Step 1.4.1.3.1

Factor $4$ out of $36 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) - 4 u}}{2 \cdot 5}$

Step 1.4.1.3.2

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

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) + 4 (- u)}}{2 \cdot 5}$

Step 1.4.1.3.3

Factor $4$ out of $4 (9 x^{2}) + 4 (- u)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - u)}}{2 \cdot 5}$

Step 1.4.1.4

Replace all occurrences of $u$ with $5 \cdot (5 x^{2} - 12)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 \cdot (5 x^{2} - 12)))}}{2 \cdot 5}$

Step 1.4.1.5

Simplify.

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Step 1.4.1.5.1

Simplify each term.

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Step 1.4.1.5.1.1

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 (5 x^{2}) + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.4.1.5.1.2

Multiply $5$ by $5$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.4.1.5.1.3

Multiply $5$ by $- 12$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} - 60))}}{2 \cdot 5}$

Step 1.4.1.5.1.4

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.4.1.5.1.5

Multiply $25$ by $- 1$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.4.1.5.1.6

Multiply $- 1$ by $- 60$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.4.1.5.2

Subtract $25 x^{2}$ from $9 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (- 16 x^{2} + 60)}}{2 \cdot 5}$

Step 1.4.1.6

Factor $4$ out of $- 16 x^{2} + 60$ .

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Step 1.4.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.4.1.6.2

Factor $4$ out of $60$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 4 (15))}}{2 \cdot 5}$

Step 1.4.1.6.3

Factor $4$ out of $4 (- 4 x^{2}) + 4 (15)$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

$y = \frac{6 x \pm \sqrt{4 \cdot (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

Step 1.4.1.7

Multiply $4$ by $4$ .

$y = \frac{6 x \pm \sqrt{16 (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.4.1.8

Rewrite $16 (- 4 x^{2} + 15)$ as ${(2^{2})}^{2} (- 4 x^{2} + 15)$ .

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Step 1.4.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{6 x \pm \sqrt{4^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.4.1.8.2

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

$y = \frac{6 x \pm \sqrt{{(2^{2})}^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{6 x \pm 2^{2} \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.4.1.10

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

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.4.2

Multiply $2$ by $5$ .

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$

Step 1.4.3

Simplify $\frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$ .

$y = \frac{3 x \pm 2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{3 x + 2 \sqrt{- 4 x^{2} + 15}}{5}$

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

Add parentheses.

$y = \frac{6 x \pm \sqrt{{(- 6 x)}^{2} - 4 \cdot (5 \cdot (5 x^{2} - 12))}}{2 \cdot 5}$

Step 1.5.1.2

Let $u = 5 \cdot (5 x^{2} - 12)$ . Substitute $u$ for all occurrences of $5 \cdot (5 x^{2} - 12)$ .

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Step 1.5.1.2.1

Apply the product rule to $- 6 x$ .

$y = \frac{6 x \pm \sqrt{{(- 6)}^{2} x^{2} - 4 \cdot u}}{2 \cdot 5}$

Step 1.5.1.2.2

Raise $- 6$ to the power of $2$ .

$y = \frac{6 x \pm \sqrt{36 x^{2} - 4 u}}{2 \cdot 5}$

Step 1.5.1.3

Factor $4$ out of $36 x^{2} - 4 u$ .

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Step 1.5.1.3.1

Factor $4$ out of $36 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) - 4 u}}{2 \cdot 5}$

Step 1.5.1.3.2

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

$y = \frac{6 x \pm \sqrt{4 (9 x^{2}) + 4 (- u)}}{2 \cdot 5}$

Step 1.5.1.3.3

Factor $4$ out of $4 (9 x^{2}) + 4 (- u)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - u)}}{2 \cdot 5}$

Step 1.5.1.4

Replace all occurrences of $u$ with $5 \cdot (5 x^{2} - 12)$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 \cdot (5 x^{2} - 12)))}}{2 \cdot 5}$

Step 1.5.1.5

Simplify.

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Step 1.5.1.5.1

Simplify each term.

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Step 1.5.1.5.1.1

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (5 (5 x^{2}) + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.5.1.5.1.2

Multiply $5$ by $5$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} + 5 \cdot - 12))}}{2 \cdot 5}$

Step 1.5.1.5.1.3

Multiply $5$ by $- 12$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2} - 60))}}{2 \cdot 5}$

Step 1.5.1.5.1.4

Apply the distributive property.

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - (25 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.5.1.5.1.5

Multiply $25$ by $- 1$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.5.1.5.1.6

Multiply $- 1$ by $- 60$ .

$y = \frac{6 x \pm \sqrt{4 (9 x^{2} - 25 x^{2} + 60)}}{2 \cdot 5}$

Step 1.5.1.5.2

Subtract $25 x^{2}$ from $9 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (- 16 x^{2} + 60)}}{2 \cdot 5}$

Step 1.5.1.6

Factor $4$ out of $- 16 x^{2} + 60$ .

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Step 1.5.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 60)}}{2 \cdot 5}$

Step 1.5.1.6.2

Factor $4$ out of $60$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2}) + 4 (15))}}{2 \cdot 5}$

Step 1.5.1.6.3

Factor $4$ out of $4 (- 4 x^{2}) + 4 (15)$ .

$y = \frac{6 x \pm \sqrt{4 (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

$y = \frac{6 x \pm \sqrt{4 \cdot (4 (- 4 x^{2} + 15))}}{2 \cdot 5}$

Step 1.5.1.7

Multiply $4$ by $4$ .

$y = \frac{6 x \pm \sqrt{16 (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.5.1.8

Rewrite $16 (- 4 x^{2} + 15)$ as ${(2^{2})}^{2} (- 4 x^{2} + 15)$ .

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Step 1.5.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{6 x \pm \sqrt{4^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.5.1.8.2

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

$y = \frac{6 x \pm \sqrt{{(2^{2})}^{2} (- 4 x^{2} + 15)}}{2 \cdot 5}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{6 x \pm 2^{2} \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.5.1.10

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

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{2 \cdot 5}$

Step 1.5.2

Multiply $2$ by $5$ .

$y = \frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$

Step 1.5.3

Simplify $\frac{6 x \pm 4 \sqrt{- 4 x^{2} + 15}}{10}$ .

$y = \frac{3 x \pm 2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{3 x - 2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{3 x + 2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x - 2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x + 2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x - 2 \sqrt{- 4 x^{2} + 15}}{5}$

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 $\frac{3 x + 2 \sqrt{- 4 x^{2} + 15}}{5}$ into two fractions.

$y = \frac{3 x}{5} + \frac{2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x - 2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 4

Split the fraction $\frac{3 x - 2 \sqrt{- 4 x^{2} + 15}}{5}$ into two fractions.

$y = \frac{3 x}{5} + \frac{2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x}{5} + \frac{- 2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 5

Move the negative in front of the fraction.

$y = \frac{3 x}{5} + \frac{2 \sqrt{- 4 x^{2} + 15}}{5}$

$y = \frac{3 x}{5} - \frac{2 \sqrt{- 4 x^{2} + 15}}{5}$

Step 6

Reorder terms.

$y = \frac{3}{5} x + \frac{2}{5} \cdot \sqrt{- 4 x^{2} + 15}$

$y = \frac{3}{5} x - (\frac{2}{5} \cdot \sqrt{- 4 x^{2} + 15})$

Step 7

Remove parentheses.

$y = \frac{3}{5} x + \frac{2}{5} \cdot \sqrt{- 4 x^{2} + 15}$

$y = \frac{3}{5} x - \frac{2}{5} \cdot \sqrt{- 4 x^{2} + 15}$

Step 8