Basic Math Examples

$y^{2} + \frac{7 y}{\sqrt{3}} - \frac{368}{3} = 0$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{7 y}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{2} + \frac{7 y}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \frac{368}{3} = 0$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{7 y}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{2} + \frac{7 y \sqrt{3}}{\sqrt{3} \sqrt{3}} - \frac{368}{3} = 0$

Step 1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$y^{2} + \frac{7 y \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \frac{368}{3} = 0$

Step 1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$y^{2} + \frac{7 y \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \frac{368}{3} = 0$

Step 1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + \frac{7 y \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \frac{368}{3} = 0$

Step 1.2.5

Add $1$ and $1$ .

$y^{2} + \frac{7 y \sqrt{3}}{{\sqrt{3}}^{2}} - \frac{368}{3} = 0$

Step 1.2.6

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

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Step 1.2.6.1

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

$y^{2} + \frac{7 y \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \frac{368}{3} = 0$

Step 1.2.6.2

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

$y^{2} + \frac{7 y \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \frac{368}{3} = 0$

Step 1.2.6.3

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

$y^{2} + \frac{7 y \sqrt{3}}{3^{\frac{2}{2}}} - \frac{368}{3} = 0$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$y^{2} + \frac{7 y \sqrt{3}}{3^{\frac{2}{2}}} - \frac{368}{3} = 0$

Step 1.2.6.4.2

Rewrite the expression.

$y^{2} + \frac{7 y \sqrt{3}}{3^{1}} - \frac{368}{3} = 0$

Step 1.2.6.5

Evaluate the exponent.

$y^{2} + \frac{7 y \sqrt{3}}{3} - \frac{368}{3} = 0$

Step 2

Multiply through by the least common denominator $3$ , then simplify.

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Step 2.1

Apply the distributive property.

$3 y^{2} + 3 (\frac{7 y \sqrt{3}}{3}) + 3 (- \frac{368}{3}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$3 y^{2} + 3 (\frac{7 y \sqrt{3}}{3}) + 3 (- \frac{368}{3}) = 0$

Step 2.2.1.2

Rewrite the expression.

$3 y^{2} + 7 y \sqrt{3} + 3 (- \frac{368}{3}) = 0$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{368}{3}$ into the numerator.

$3 y^{2} + 7 y \sqrt{3} + 3 (\frac{- 368}{3}) = 0$

Step 2.2.2.2

Cancel the common factor.

$3 y^{2} + 7 y \sqrt{3} + 3 (\frac{- 368}{3}) = 0$

Step 2.2.2.3

Rewrite the expression.

$3 y^{2} + 7 y \sqrt{3} - 368 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = 7 \sqrt{3}$ , and $c = - 368$ into the quadratic formula and solve for $y$ .

$\frac{- 7 \sqrt{3} \pm \sqrt{{(7 \sqrt{3})}^{2} - 4 \cdot (3 \cdot - 368)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the product rule to $7 \sqrt{3}$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{7^{2} {\sqrt{3}}^{2} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.2

Raise $7$ to the power of $2$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 {\sqrt{3}}^{2} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3

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

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Step 5.1.3.1

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

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 {(3^{\frac{1}{2}})}^{2} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3.2

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

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 \cdot 3^{\frac{1}{2} \cdot 2} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3.3

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

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 \cdot 3^{\frac{2}{2}} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3.4

Cancel the common factor of $2$ .

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Step 5.1.3.4.1

Cancel the common factor.

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 \cdot 3^{\frac{2}{2}} - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3.4.2

Rewrite the expression.

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 \cdot 3 - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.3.5

Evaluate the exponent.

$y = \frac{- 7 \sqrt{3} \pm \sqrt{49 \cdot 3 - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.4

Multiply $49$ by $3$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{147 - 4 \cdot 3 \cdot - 368}}{2 \cdot 3}$

Step 5.1.5

Multiply $- 4 \cdot 3 \cdot - 368$ .

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Step 5.1.5.1

Multiply $- 4$ by $3$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{147 - 12 \cdot - 368}}{2 \cdot 3}$

Step 5.1.5.2

Multiply $- 12$ by $- 368$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{147 + 4416}}{2 \cdot 3}$

Step 5.1.6

Add $147$ and $4416$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{4563}}{2 \cdot 3}$

Step 5.1.7

Rewrite $4563$ as $39^{2} \cdot 3$ .

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Step 5.1.7.1

Factor $1521$ out of $4563$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{1521 (3)}}{2 \cdot 3}$

Step 5.1.7.2

Rewrite $1521$ as $39^{2}$ .

$y = \frac{- 7 \sqrt{3} \pm \sqrt{39^{2} \cdot 3}}{2 \cdot 3}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{- 7 \sqrt{3} \pm 39 \sqrt{3}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$y = \frac{- 7 \sqrt{3} \pm 39 \sqrt{3}}{6}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{16 \sqrt{3}}{3}, - \frac{23 \sqrt{3}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{16 \sqrt{3}}{3}, - \frac{23 \sqrt{3}}{3}$

Decimal Form:

$y = 9.23760430 \dots, - 13.27905619 \dots$