Trigonometry Examples

$7 x^{2} + 6 \sqrt{3} x y + 13 y^{2} - 6 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 7$ , $b = 6 \sqrt{3} y$ , and $c = 13 y^{2} - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \sqrt{3} y \pm \sqrt{{(6 \sqrt{3} y)}^{2} - 4 \cdot (7 \cdot (13 y^{2} - 6))}}{2 \cdot 7}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add parentheses.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{{(6 (\sqrt{3} y))}^{2} - 4 \cdot (7 \cdot (13 y^{2} - 6))}}{2 \cdot 7}$

Step 3.1.2

Let $u = 7 \cdot (13 y^{2} - 6)$ . Substitute $u$ for all occurrences of $7 \cdot (13 y^{2} - 6)$ .

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Step 3.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.1.1

Apply the product rule to $6 \sqrt{3} y$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{{(6 \sqrt{3})}^{2} y^{2} - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.1.2

Apply the product rule to $6 \sqrt{3}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{6^{2} {\sqrt{3}}^{2} y^{2} - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.2

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

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 {\sqrt{3}}^{2} y^{2} - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 {(3^{\frac{1}{2}})}^{2} y^{2} - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 \cdot (3^{\frac{1}{2} \cdot 2} y^{2}) - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 \cdot (3^{\frac{2}{2}} y^{2}) - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.3.4.1

Cancel the common factor.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 \cdot (3^{\frac{2}{2}} y^{2}) - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3.4.2

Rewrite the expression.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 \cdot (3 y^{2}) - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.3.5

Evaluate the exponent.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{36 \cdot (3 y^{2}) - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.4

Multiply $36$ by $3$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{108 y^{2} - 4 u}}{2 \cdot 7}$

Step 3.1.3

Factor $4$ out of $108 y^{2} - 4 u$ .

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Step 3.1.3.1

Factor $4$ out of $108 y^{2}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2}) - 4 u}}{2 \cdot 7}$

Step 3.1.3.2

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

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2}) + 4 (- u)}}{2 \cdot 7}$

Step 3.1.3.3

Factor $4$ out of $4 (27 y^{2}) + 4 (- u)$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - u)}}{2 \cdot 7}$

Step 3.1.4

Replace all occurrences of $u$ with $7 \cdot (13 y^{2} - 6)$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - (7 \cdot (13 y^{2} - 6)))}}{2 \cdot 7}$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

Apply the distributive property.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - (7 (13 y^{2}) + 7 \cdot - 6))}}{2 \cdot 7}$

Step 3.1.5.1.2

Multiply $13$ by $7$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - (91 y^{2} + 7 \cdot - 6))}}{2 \cdot 7}$

Step 3.1.5.1.3

Multiply $7$ by $- 6$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - (91 y^{2} - 42))}}{2 \cdot 7}$

Step 3.1.5.1.4

Apply the distributive property.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - (91 y^{2}) + 42)}}{2 \cdot 7}$

Step 3.1.5.1.5

Multiply $91$ by $- 1$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - 91 y^{2} + 42)}}{2 \cdot 7}$

Step 3.1.5.1.6

Multiply $- 1$ by $- 42$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (27 y^{2} - 91 y^{2} + 42)}}{2 \cdot 7}$

Step 3.1.5.2

Subtract $91 y^{2}$ from $27 y^{2}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (- 64 y^{2} + 42)}}{2 \cdot 7}$

Step 3.1.6

Factor $2$ out of $- 64 y^{2} + 42$ .

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Step 3.1.6.1

Factor $2$ out of $- 64 y^{2}$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (2 (- 32 y^{2}) + 42)}}{2 \cdot 7}$

Step 3.1.6.2

Factor $2$ out of $42$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (2 (- 32 y^{2}) + 2 (21))}}{2 \cdot 7}$

Step 3.1.6.3

Factor $2$ out of $2 (- 32 y^{2}) + 2 (21)$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (2 (- 32 y^{2} + 21))}}{2 \cdot 7}$

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 \cdot (2 (- 32 y^{2} + 21))}}{2 \cdot 7}$

Step 3.1.7

Multiply $4$ by $2$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{8 (- 32 y^{2} + 21)}}{2 \cdot 7}$

Step 3.1.8

Rewrite $8 (- 32 y^{2} + 21)$ as $2^{2} \cdot (2 (- 32 y^{2} + 21))$ .

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Step 3.1.8.1

Factor $4$ out of $8$ .

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{4 (2) (- 32 y^{2} + 21)}}{2 \cdot 7}$

Step 3.1.8.2

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

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{2^{2} \cdot (2 (- 32 y^{2} + 21))}}{2 \cdot 7}$

Step 3.1.8.3

Add parentheses.

$x = \frac{- 6 \sqrt{3} y \pm \sqrt{2^{2} \cdot (2 (- 32 y^{2} + 21))}}{2 \cdot 7}$

Step 3.1.9

Pull terms out from under the radical.

$x = \frac{- 6 \sqrt{3} y \pm 2 \sqrt{2 (- 32 y^{2} + 21)}}{2 \cdot 7}$

Step 3.2

Multiply $2$ by $7$ .

$x = \frac{- 6 \sqrt{3} y \pm 2 \sqrt{2 (- 32 y^{2} + 21)}}{14}$

Step 3.3

Simplify $\frac{- 6 \sqrt{3} y \pm 2 \sqrt{2 (- 32 y^{2} + 21)}}{14}$ .

$x = \frac{- 3 \sqrt{3} y \pm \sqrt{2 (- 32 y^{2} + 21)}}{7}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{3 \sqrt{3} y - \sqrt{2 (- 32 y^{2} + 21)}}{7}$

$x = - \frac{3 \sqrt{3} y + \sqrt{2 (- 32 y^{2} + 21)}}{7}$