Precalculus Examples

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

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

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

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.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

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

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Step 1.3.1.2.1.1

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

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

Step 1.3.1.2.1.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{10^{2} {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3

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

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Step 1.3.1.2.3.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {(3^{\frac{1}{2}})}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{1}{2} \cdot 2} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3.3

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3.4

Cancel the common factor of $2$ .

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Step 1.3.1.2.3.4.1

Cancel the common factor.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3.4.2

Rewrite the expression.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.3.5

Evaluate the exponent.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.3.1.2.4

Multiply $100$ by $3$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{300 x^{2} - 4 u}}{2 \cdot 21}$

Step 1.3.1.3

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

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Step 1.3.1.3.1

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

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

Step 1.3.1.3.2

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

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

Step 1.3.1.3.3

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

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

Step 1.3.1.4

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

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

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.

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

Step 1.3.1.5.1.2

Multiply $31$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} + 21 \cdot - 144))}}{2 \cdot 21}$

Step 1.3.1.5.1.3

Multiply $21$ by $- 144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} - 3024))}}{2 \cdot 21}$

Step 1.3.1.5.1.4

Apply the distributive property.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.3.1.5.1.5

Multiply $651$ by $- 1$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.3.1.5.1.6

Multiply $- 1$ by $- 3024$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.3.1.5.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (- 576 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.3.1.6

Factor $144$ out of $- 576 x^{2} + 3024$ .

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Step 1.3.1.6.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (144 (- 4 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.3.1.6.2

Factor $144$ out of $3024$ .

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

Step 1.3.1.6.3

Factor $144$ out of $144 (- 4 x^{2}) + 144 (21)$ .

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

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

Step 1.3.1.7

Multiply $4$ by $144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{576 (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.3.1.8

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{24^{2} (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{2 \cdot 21}$

Step 1.3.2

Multiply $2$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$

Step 1.3.3

Simplify $\frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$ .

$y = \frac{- 5 \sqrt{3} x \pm 12 \sqrt{- 4 x^{2} + 21}}{21}$

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.

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

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

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Step 1.4.1.2.1.1

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

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

Step 1.4.1.2.1.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{10^{2} {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3

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

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Step 1.4.1.2.3.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {(3^{\frac{1}{2}})}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{1}{2} \cdot 2} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3.3

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.3.4.1

Cancel the common factor.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3.4.2

Rewrite the expression.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.3.5

Evaluate the exponent.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.4.1.2.4

Multiply $100$ by $3$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{300 x^{2} - 4 u}}{2 \cdot 21}$

Step 1.4.1.3

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

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Step 1.4.1.3.1

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

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

Step 1.4.1.3.2

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

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

Step 1.4.1.3.3

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

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

Step 1.4.1.4

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

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

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.

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

Step 1.4.1.5.1.2

Multiply $31$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} + 21 \cdot - 144))}}{2 \cdot 21}$

Step 1.4.1.5.1.3

Multiply $21$ by $- 144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} - 3024))}}{2 \cdot 21}$

Step 1.4.1.5.1.4

Apply the distributive property.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.4.1.5.1.5

Multiply $651$ by $- 1$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.4.1.5.1.6

Multiply $- 1$ by $- 3024$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.4.1.5.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (- 576 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.4.1.6

Factor $144$ out of $- 576 x^{2} + 3024$ .

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Step 1.4.1.6.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (144 (- 4 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.4.1.6.2

Factor $144$ out of $3024$ .

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

Step 1.4.1.6.3

Factor $144$ out of $144 (- 4 x^{2}) + 144 (21)$ .

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

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

Step 1.4.1.7

Multiply $4$ by $144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{576 (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.4.1.8

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{24^{2} (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{2 \cdot 21}$

Step 1.4.2

Multiply $2$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$

Step 1.4.3

Simplify $\frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$ .

$y = \frac{- 5 \sqrt{3} x \pm 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{- 5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.4.5

Factor $- 1$ out of $- 5 \sqrt{3} x$ .

$y = \frac{- (5 \sqrt{3} x) + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.4.6

Factor $- 1$ out of $12 \sqrt{- 4 x^{2} + 21}$ .

$y = \frac{- (5 \sqrt{3} x) - (- 12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.4.7

Factor $- 1$ out of $- (5 \sqrt{3} x) - (- 12 \sqrt{- 4 x^{2} + 21})$ .

$y = \frac{- (5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.4.8

Rewrite $- (5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21})$ as $- 1 (5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21})$ .

$y = \frac{- 1 (5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.4.9

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21}}{21}$

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.

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

Step 1.5.1.2

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

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Step 1.5.1.2.1

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

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Step 1.5.1.2.1.1

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

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

Step 1.5.1.2.1.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{10^{2} {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {\sqrt{3}}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3

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

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Step 1.5.1.2.3.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 {(3^{\frac{1}{2}})}^{2} x^{2} - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{1}{2} \cdot 2} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3.3

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3.4

Cancel the common factor of $2$ .

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Step 1.5.1.2.3.4.1

Cancel the common factor.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3^{\frac{2}{2}} x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3.4.2

Rewrite the expression.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.3.5

Evaluate the exponent.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{100 \cdot (3 x^{2}) - 4 \cdot u}}{2 \cdot 21}$

Step 1.5.1.2.4

Multiply $100$ by $3$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{300 x^{2} - 4 u}}{2 \cdot 21}$

Step 1.5.1.3

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

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Step 1.5.1.3.1

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

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

Step 1.5.1.3.2

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

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

Step 1.5.1.3.3

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

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

Step 1.5.1.4

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

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

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.

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

Step 1.5.1.5.1.2

Multiply $31$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} + 21 \cdot - 144))}}{2 \cdot 21}$

Step 1.5.1.5.1.3

Multiply $21$ by $- 144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2} - 3024))}}{2 \cdot 21}$

Step 1.5.1.5.1.4

Apply the distributive property.

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - (651 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.5.1.5.1.5

Multiply $651$ by $- 1$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.5.1.5.1.6

Multiply $- 1$ by $- 3024$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (75 x^{2} - 651 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.5.1.5.2

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (- 576 x^{2} + 3024)}}{2 \cdot 21}$

Step 1.5.1.6

Factor $144$ out of $- 576 x^{2} + 3024$ .

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Step 1.5.1.6.1

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{4 (144 (- 4 x^{2}) + 3024)}}{2 \cdot 21}$

Step 1.5.1.6.2

Factor $144$ out of $3024$ .

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

Step 1.5.1.6.3

Factor $144$ out of $144 (- 4 x^{2}) + 144 (21)$ .

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

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

Step 1.5.1.7

Multiply $4$ by $144$ .

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{576 (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.5.1.8

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

$y = \frac{- 10 \sqrt{3} x \pm \sqrt{24^{2} (- 4 x^{2} + 21)}}{2 \cdot 21}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{2 \cdot 21}$

Step 1.5.2

Multiply $2$ by $21$ .

$y = \frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$

Step 1.5.3

Simplify $\frac{- 10 \sqrt{3} x \pm 24 \sqrt{- 4 x^{2} + 21}}{42}$ .

$y = \frac{- 5 \sqrt{3} x \pm 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{- 5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.5.5

Factor $- 1$ out of $- 5 \sqrt{3} x$ .

$y = \frac{- (5 \sqrt{3} x) - 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.5.6

Factor $- 1$ out of $- 12 \sqrt{- 4 x^{2} + 21}$ .

$y = \frac{- (5 \sqrt{3} x) - (12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.5.7

Factor $- 1$ out of $- (5 \sqrt{3} x) - (12 \sqrt{- 4 x^{2} + 21})$ .

$y = \frac{- (5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.5.8

Rewrite $- (5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21})$ as $- 1 (5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21})$ .

$y = \frac{- 1 (5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21})}{21}$

Step 1.5.9

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 1.6

The final answer is the combination of both solutions.

$y = - \frac{5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - \frac{5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

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 $\frac{5 \sqrt{3} x - 12 \sqrt{- 4 x^{2} + 21}}{21}$ into two fractions.

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{- 12 \sqrt{- 4 x^{2} + 21}}{21})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $- 12$ and $21$ .

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Step 4.1.1

Factor $3$ out of $- 12 \sqrt{- 4 x^{2} + 21}$ .

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{3 (- 4 \sqrt{- 4 x^{2} + 21})}{21})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $3$ out of $21$ .

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{3 (- 4 \sqrt{- 4 x^{2} + 21})}{3 (7)})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 4.1.2.2

Cancel the common factor.

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{3 (- 4 \sqrt{- 4 x^{2} + 21})}{3 \cdot 7}), - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 4.1.2.3

Rewrite the expression.

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{- 4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{- 4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{- 4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 4.2

Move the negative in front of the fraction.

$y = - (\frac{5 \sqrt{3} x}{21} - \frac{4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

$y = - (\frac{5 \sqrt{3} x}{21} - \frac{4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 5

Apply the distributive property.

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - \frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$

Step 6

Split the fraction $\frac{5 \sqrt{3} x + 12 \sqrt{- 4 x^{2} + 21}}{21}$ into two fractions.

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{12 \sqrt{- 4 x^{2} + 21}}{21})$

Step 7

Cancel the common factor of $12$ and $21$ .

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Step 7.1

Factor $3$ out of $12 \sqrt{- 4 x^{2} + 21}$ .

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{3 (4 \sqrt{- 4 x^{2} + 21})}{21})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $21$ .

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{3 (4 \sqrt{- 4 x^{2} + 21})}{3 (7)})$

Step 7.2.2

Cancel the common factor.

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}, - (\frac{5 \sqrt{3} x}{21} + \frac{3 (4 \sqrt{- 4 x^{2} + 21})}{3 \cdot 7})$

Step 7.2.3

Rewrite the expression.

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7})$

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - (\frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7})$

Step 8

Apply the distributive property.

$y = - \frac{5 \sqrt{3} x}{21} + \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

$y = - \frac{5 \sqrt{3} x}{21} - \frac{4 \sqrt{- 4 x^{2} + 21}}{7}$

Step 9

Reorder terms.

$y = - (\frac{5 \sqrt{3}}{21} x) + \frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21}$

$y = - (\frac{5 \sqrt{3}}{21} x) - (\frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21})$

Step 10

Remove parentheses.

$y = - \frac{5 \sqrt{3}}{21} x + \frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21}$

$y = - (\frac{5 \sqrt{3}}{21} x) - (\frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21})$

Step 11

Remove parentheses.

$y = - \frac{5 \sqrt{3}}{21} x + \frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21}$

$y = - \frac{5 \sqrt{3}}{21} x - (\frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21})$

Step 12

Remove parentheses.

$y = - \frac{5 \sqrt{3}}{21} x + \frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21}$

$y = - \frac{5 \sqrt{3}}{21} x - \frac{4}{7} \cdot \sqrt{- 4 x^{2} + 21}$

Step 13