Precalculus Examples

$31 x^{2} + 10 \sqrt{3} x y + 21 y^{2} - 144 = 0$

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 $a = 31$ , $b = 10 \sqrt{3} y$ , and $c = 21 y^{2} - 144$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \sqrt{3} y \pm \sqrt{{(10 \sqrt{3} y)}^{2} - 4 \cdot (31 \cdot (21 y^{2} - 144))}}{2 \cdot 31}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add parentheses.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{{(10 (\sqrt{3} y))}^{2} - 4 \cdot (31 \cdot (21 y^{2} - 144))}}{2 \cdot 31}$

Step 3.1.2

Let $u = 31 \cdot (21 y^{2} - 144)$ . Substitute $u$ for all occurrences of $31 \cdot (21 y^{2} - 144)$ .

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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 $10 \sqrt{3} y$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{{(10 \sqrt{3})}^{2} y^{2} - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.1.2

Apply the product rule to $10 \sqrt{3}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{10^{2} {\sqrt{3}}^{2} y^{2} - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.2

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 {\sqrt{3}}^{2} y^{2} - 4 \cdot u}}{2 \cdot 31}$

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 $3^{\frac{1}{2}}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 {(3^{\frac{1}{2}})}^{2} y^{2} - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.3.2

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

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 \cdot (3^{\frac{1}{2} \cdot 2} y^{2}) - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.3.3

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

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 \cdot (3^{\frac{2}{2}} y^{2}) - 4 \cdot u}}{2 \cdot 31}$

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.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 \cdot (3^{\frac{2}{2}} y^{2}) - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.3.4.2

Rewrite the expression.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 \cdot (3 y^{2}) - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.3.5

Evaluate the exponent.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{100 \cdot (3 y^{2}) - 4 \cdot u}}{2 \cdot 31}$

Step 3.1.2.4

Multiply $100$ by $3$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{300 y^{2} - 4 u}}{2 \cdot 31}$

Step 3.1.3

Factor $4$ out of $300 y^{2} - 4 u$ .

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Step 3.1.3.1

Factor $4$ out of $300 y^{2}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2}) - 4 u}}{2 \cdot 31}$

Step 3.1.3.2

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

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2}) + 4 (- u)}}{2 \cdot 31}$

Step 3.1.3.3

Factor $4$ out of $4 (75 y^{2}) + 4 (- u)$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - u)}}{2 \cdot 31}$

Step 3.1.4

Replace all occurrences of $u$ with $31 \cdot (21 y^{2} - 144)$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - (31 \cdot (21 y^{2} - 144)))}}{2 \cdot 31}$

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.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - (31 (21 y^{2}) + 31 \cdot - 144))}}{2 \cdot 31}$

Step 3.1.5.1.2

Multiply $21$ by $31$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - (651 y^{2} + 31 \cdot - 144))}}{2 \cdot 31}$

Step 3.1.5.1.3

Multiply $31$ by $- 144$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - (651 y^{2} - 4464))}}{2 \cdot 31}$

Step 3.1.5.1.4

Apply the distributive property.

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - (651 y^{2}) + 4464)}}{2 \cdot 31}$

Step 3.1.5.1.5

Multiply $651$ by $- 1$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - 651 y^{2} + 4464)}}{2 \cdot 31}$

Step 3.1.5.1.6

Multiply $- 1$ by $- 4464$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (75 y^{2} - 651 y^{2} + 4464)}}{2 \cdot 31}$

Step 3.1.5.2

Subtract $651 y^{2}$ from $75 y^{2}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (- 576 y^{2} + 4464)}}{2 \cdot 31}$

Step 3.1.6

Factor $144$ out of $- 576 y^{2} + 4464$ .

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Step 3.1.6.1

Factor $144$ out of $- 576 y^{2}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (144 (- 4 y^{2}) + 4464)}}{2 \cdot 31}$

Step 3.1.6.2

Factor $144$ out of $4464$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (144 (- 4 y^{2}) + 144 (31))}}{2 \cdot 31}$

Step 3.1.6.3

Factor $144$ out of $144 (- 4 y^{2}) + 144 (31)$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 (144 (- 4 y^{2} + 31))}}{2 \cdot 31}$

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{4 \cdot (144 (- 4 y^{2} + 31))}}{2 \cdot 31}$

Step 3.1.7

Multiply $4$ by $144$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{576 (- 4 y^{2} + 31)}}{2 \cdot 31}$

Step 3.1.8

Rewrite $576$ as $24^{2}$ .

$x = \frac{- 10 \sqrt{3} y \pm \sqrt{24^{2} (- 4 y^{2} + 31)}}{2 \cdot 31}$

Step 3.1.9

Pull terms out from under the radical.

$x = \frac{- 10 \sqrt{3} y \pm 24 \sqrt{- 4 y^{2} + 31}}{2 \cdot 31}$

Step 3.2

Multiply $2$ by $31$ .

$x = \frac{- 10 \sqrt{3} y \pm 24 \sqrt{- 4 y^{2} + 31}}{62}$

Step 3.3

Simplify $\frac{- 10 \sqrt{3} y \pm 24 \sqrt{- 4 y^{2} + 31}}{62}$ .

$x = \frac{- 5 \sqrt{3} y \pm 12 \sqrt{- 4 y^{2} + 31}}{31}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{5 \sqrt{3} y - 12 \sqrt{- 4 y^{2} + 31}}{31}$

$x = - \frac{5 \sqrt{3} y + 12 \sqrt{- 4 y^{2} + 31}}{31}$