Basic Math Examples

$3 \cdot \sqrt{a} + a \cdot \sqrt{3} + a - 3 = 0$

Step 1

Move all terms not containing $3 \cdot \sqrt{a}$ to the right side of the equation.

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Step 1.1

Subtract $a \sqrt{3}$ from both sides of the equation.

$3 \cdot \sqrt{a} + a - 3 = - a \sqrt{3}$

Step 1.2

Subtract $a$ from both sides of the equation.

$3 \cdot \sqrt{a} - 3 = - a \sqrt{3} - a$

Step 1.3

Add $3$ to both sides of the equation.

$3 \cdot \sqrt{a} = - a \sqrt{3} - a + 3$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(3 \cdot \sqrt{a})}^{2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(3 \cdot a^{\frac{1}{2}})}^{2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(3 \cdot a^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $3 a^{\frac{1}{2}}$ .

$3^{2} {(a^{\frac{1}{2}})}^{2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2.1.2

Raise $3$ to the power of $2$ .

$9 {(a^{\frac{1}{2}})}^{2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2.1.3

Multiply the exponents in ${(a^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

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

$9 a^{\frac{1}{2} \cdot 2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$9 a^{\frac{1}{2} \cdot 2} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$9 a^{1} = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.2.1.4

Simplify.

$9 a = {(- a \sqrt{3} - a + 3)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- a \sqrt{3} - a + 3)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- a \sqrt{3} - a + 3)}^{2}$ as $(- a \sqrt{3} - a + 3) (- a \sqrt{3} - a + 3)$ .

$9 a = (- a \sqrt{3} - a + 3) (- a \sqrt{3} - a + 3)$

Step 3.3.1.2

Expand $(- a \sqrt{3} - a + 3) (- a \sqrt{3} - a + 3)$ by multiplying each term in the first expression by each term in the second expression.

$9 a = - a \sqrt{3} (- a \sqrt{3}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3

Simplify terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 3.3.1.3.1.1.1

Move $a$ .

$9 a = - (a \cdot a) \sqrt{3} (- 1 \sqrt{3}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.1.2

Multiply $a$ by $a$ .

$9 a = - a^{2} \sqrt{3} (- 1 \sqrt{3}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$9 a = - a^{2} \sqrt{3} (- \sqrt{3}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3

Multiply $- a^{2} \sqrt{3} (- \sqrt{3})$ .

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Step 3.3.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$9 a = 1 a^{2} \sqrt{3} \sqrt{3} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3.2

Multiply $a^{2}$ by $1$ .

$9 a = a^{2} \sqrt{3} \sqrt{3} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3.3

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

$9 a = a^{2} ({\sqrt{3}}^{1} \sqrt{3}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3.4

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

$9 a = a^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3.5

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

$9 a = a^{2} {\sqrt{3}}^{1 + 1} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.3.6

Add $1$ and $1$ .

$9 a = a^{2} {\sqrt{3}}^{2} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$9 a = a^{2} {(3^{\frac{1}{2}})}^{2} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4.2

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

$9 a = a^{2} \cdot 3^{\frac{1}{2} \cdot 2} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4.3

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

$9 a = a^{2} \cdot 3^{\frac{2}{2}} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$9 a = a^{2} \cdot 3^{\frac{2}{2}} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$9 a = a^{2} \cdot 3^{1} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.4.5

Evaluate the exponent.

$9 a = a^{2} \cdot 3 - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.5

Move $3$ to the left of $a^{2}$ .

$9 a = 3 \cdot a^{2} - a \sqrt{3} (- a) - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.6

Multiply $a$ by $a$ by adding the exponents.

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Step 3.3.1.3.1.6.1

Move $a$ .

$9 a = 3 a^{2} - (a \cdot a) \sqrt{3} \cdot - 1 - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.6.2

Multiply $a$ by $a$ .

$9 a = 3 a^{2} - a^{2} \sqrt{3} \cdot - 1 - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.7

Multiply $- a^{2} \sqrt{3} \cdot - 1$ .

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Step 3.3.1.3.1.7.1

Multiply $- 1$ by $- 1$ .

$9 a = 3 a^{2} + 1 a^{2} \sqrt{3} - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.7.2

Multiply $a^{2}$ by $1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - a \sqrt{3} \cdot 3 - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.8

Multiply $3$ by $- 1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} - a (- a \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.9

Multiply $a$ by $a$ by adding the exponents.

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Step 3.3.1.3.1.9.1

Move $a$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} - (a \cdot a) (- 1 \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.9.2

Multiply $a$ by $a$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} - a^{2} (- 1 \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.10

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} - a^{2} (- \sqrt{3}) - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.11

Multiply $- a^{2} (- \sqrt{3})$ .

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Step 3.3.1.3.1.11.1

Multiply $- 1$ by $- 1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + 1 a^{2} \sqrt{3} - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.11.2

Multiply $a^{2}$ by $1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} - a (- a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.12

Rewrite using the commutative property of multiplication.

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} - 1 \cdot - 1 a \cdot a - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.13

Multiply $a$ by $a$ by adding the exponents.

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Step 3.3.1.3.1.13.1

Move $a$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} - 1 \cdot - 1 (a \cdot a) - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.13.2

Multiply $a$ by $a$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} - 1 \cdot - 1 a^{2} - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.14

Multiply $- 1$ by $- 1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + 1 a^{2} - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.15

Multiply $a^{2}$ by $1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + a^{2} - a \cdot 3 + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.16

Multiply $3$ by $- 1$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + a^{2} - 3 a + 3 (- a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.17

Multiply $- 1$ by $3$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + a^{2} - 3 a - 3 (a \sqrt{3}) + 3 (- a) + 3 \cdot 3$

Step 3.3.1.3.1.18

Multiply $- 1$ by $3$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + a^{2} - 3 a - 3 a \sqrt{3} - 3 a + 3 \cdot 3$

Step 3.3.1.3.1.19

Multiply $3$ by $3$ .

$9 a = 3 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} + a^{2} - 3 a - 3 a \sqrt{3} - 3 a + 9$

Step 3.3.1.3.2

Simplify by adding terms.

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Step 3.3.1.3.2.1

Add $3 a^{2}$ and $a^{2}$ .

$9 a = 4 a^{2} + a^{2} \sqrt{3} - 3 a \sqrt{3} + a^{2} \sqrt{3} - 3 a - 3 a \sqrt{3} - 3 a + 9$

Step 3.3.1.3.2.2

Add $a^{2} \sqrt{3}$ and $a^{2} \sqrt{3}$ .

$9 a = 4 a^{2} + 2 a^{2} \sqrt{3} - 3 a \sqrt{3} - 3 a - 3 a \sqrt{3} - 3 a + 9$

Step 3.3.1.3.2.3

Subtract $3 a \sqrt{3}$ from $- 3 a \sqrt{3}$ .

$9 a = 4 a^{2} + 2 a^{2} \sqrt{3} - 3 a - 6 a \sqrt{3} - 3 a + 9$

Step 3.3.1.3.2.4

Subtract $3 a$ from $- 3 a$ .

$9 a = 4 a^{2} + 2 a^{2} \sqrt{3} - 6 a - 6 a \sqrt{3} + 9$

Step 4

Solve for $a$ .

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Step 4.1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 a^{2} + 2 a^{2} \sqrt{3} - 6 a - 6 a \sqrt{3} + 9 = 9 a$

Step 4.2

Move all terms containing $a$ to the left side of the equation.

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Step 4.2.1

Subtract $9 a$ from both sides of the equation.

$4 a^{2} + 2 a^{2} \sqrt{3} - 6 a - 6 a \sqrt{3} + 9 - 9 a = 0$

Step 4.2.2

Subtract $9 a$ from $- 6 a$ .

$4 a^{2} + 2 a^{2} \sqrt{3} - 15 a - 6 a \sqrt{3} + 9 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Regroup terms.

$2 a^{2} \sqrt{3} - 6 a \sqrt{3} + 4 a^{2} - 15 a + 9 = 0$

Step 4.3.2

Factor $2 a \sqrt{3}$ out of $2 a^{2} \sqrt{3} - 6 a \sqrt{3}$ .

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Step 4.3.2.1

Factor $2 a \sqrt{3}$ out of $2 a^{2} \sqrt{3}$ .

$2 a \sqrt{3} (a) - 6 a \sqrt{3} + 4 a^{2} - 15 a + 9 = 0$

Step 4.3.2.2

Factor $2 a \sqrt{3}$ out of $- 6 a \sqrt{3}$ .

$2 a \sqrt{3} (a) + 2 a \sqrt{3} (- 3) + 4 a^{2} - 15 a + 9 = 0$

Step 4.3.2.3

Factor $2 a \sqrt{3}$ out of $2 a \sqrt{3} (a) + 2 a \sqrt{3} (- 3)$ .

$2 a \sqrt{3} (a - 3) + 4 a^{2} - 15 a + 9 = 0$

Step 4.3.3

Factor by grouping.

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Step 4.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 9 = 36$ and whose sum is $b = - 15$ .

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Step 4.3.3.1.1

Factor $- 15$ out of $- 15 a$ .

$2 a \sqrt{3} (a - 3) + 4 a^{2} - 15 a + 9 = 0$

Step 4.3.3.1.2

Rewrite $- 15$ as $- 3$ plus $- 12$

$2 a \sqrt{3} (a - 3) + 4 a^{2} + (- 3 - 12) a + 9 = 0$

Step 4.3.3.1.3

Apply the distributive property.

$2 a \sqrt{3} (a - 3) + 4 a^{2} - 3 a - 12 a + 9 = 0$

Step 4.3.3.2

Factor out the greatest common factor from each group.

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Step 4.3.3.2.1

Group the first two terms and the last two terms.

$2 a \sqrt{3} (a - 3) + 4 a^{2} - 3 a - 12 a + 9 = 0$

Step 4.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$2 a \sqrt{3} (a - 3) + a (4 a - 3) - 3 (4 a - 3) = 0$

Step 4.3.3.3

Factor the polynomial by factoring out the greatest common factor, $4 a - 3$ .

$2 a \sqrt{3} (a - 3) + (4 a - 3) (a - 3) = 0$

Step 4.3.4

Factor $a - 3$ out of $2 a \sqrt{3} (a - 3) + (4 a - 3) (a - 3)$ .

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Step 4.3.4.1

Factor $a - 3$ out of $2 a \sqrt{3} (a - 3)$ .

$(a - 3) (2 a \sqrt{3}) + (4 a - 3) (a - 3) = 0$

Step 4.3.4.2

Factor $a - 3$ out of $(4 a - 3) (a - 3)$ .

$(a - 3) (2 a \sqrt{3}) + (a - 3) (4 a - 3) = 0$

Step 4.3.4.3

Factor $a - 3$ out of $(a - 3) (2 a \sqrt{3}) + (a - 3) (4 a - 3)$ .

$(a - 3) (2 a \sqrt{3} + 4 a - 3) = 0$

Step 4.3.5

Reorder terms.

$(a - 3) (4 a + 2 a \sqrt{3} - 3) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a - 3 = 0$

$4 a + 2 a \sqrt{3} - 3 = 0$

Step 4.5

Set $a - 3$ equal to $0$ and solve for $a$ .

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Step 4.5.1

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Step 4.5.2

Add $3$ to both sides of the equation.

$a = 3$

Step 4.6

Set $4 a + 2 a \sqrt{3} - 3$ equal to $0$ and solve for $a$ .

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Step 4.6.1

Set $4 a + 2 a \sqrt{3} - 3$ equal to $0$ .

$4 a + 2 a \sqrt{3} - 3 = 0$

Step 4.6.2

Solve $4 a + 2 a \sqrt{3} - 3 = 0$ for $a$ .

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Step 4.6.2.1

Add $3$ to both sides of the equation.

$4 a + 2 a \sqrt{3} = 3$

Step 4.6.2.2

Factor $2 a$ out of $4 a + 2 a \sqrt{3}$ .

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Step 4.6.2.2.1

Factor $2 a$ out of $4 a$ .

$2 a (2) + 2 a \sqrt{3} = 3$

Step 4.6.2.2.2

Factor $2 a$ out of $2 a \sqrt{3}$ .

$2 a (2) + 2 a (\sqrt{3}) = 3$

Step 4.6.2.2.3

Factor $2 a$ out of $2 a (2) + 2 a (\sqrt{3})$ .

$2 a (2 + \sqrt{3}) = 3$

Step 4.6.2.3

Divide each term in $2 a (2 + \sqrt{3}) = 3$ by $4 + 2 \sqrt{3}$ and simplify.

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Step 4.6.2.3.1

Divide each term in $2 a (2 + \sqrt{3}) = 3$ by $4 + 2 \sqrt{3}$ .

$\frac{2 a (2 + \sqrt{3})}{4 + 2 \sqrt{3}} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2

Simplify the left side.

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Step 4.6.2.3.2.1

Cancel the common factor of $2$ and $4 + 2 \sqrt{3}$ .

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Step 4.6.2.3.2.1.1

Factor $2$ out of $2 a (2 + \sqrt{3})$ .

$\frac{2 (a (2 + \sqrt{3}))}{4 + 2 \sqrt{3}} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.1.2

Cancel the common factors.

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Step 4.6.2.3.2.1.2.1

Factor $2$ out of $4$ .

$\frac{2 (a (2 + \sqrt{3}))}{2 \cdot 2 + 2 \sqrt{3}} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.1.2.2

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (a (2 + \sqrt{3}))}{2 \cdot 2 + 2 (\sqrt{3})} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.1.2.3

Factor $2$ out of $2 \cdot 2 + 2 (\sqrt{3})$ .

$\frac{2 (a (2 + \sqrt{3}))}{2 \cdot (2 + \sqrt{3})} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.1.2.4

Cancel the common factor.

$\frac{2 (a (2 + \sqrt{3}))}{2 \cdot (2 + \sqrt{3})} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.1.2.5

Rewrite the expression.

$\frac{a (2 + \sqrt{3})}{2 + \sqrt{3}} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.2

Cancel the common factor of $2 + \sqrt{3}$ .

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Step 4.6.2.3.2.2.1

Cancel the common factor.

$\frac{a (2 + \sqrt{3})}{2 + \sqrt{3}} = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.2.2.2

Divide $a$ by $1$ .

$a = \frac{3}{4 + 2 \sqrt{3}}$

Step 4.6.2.3.3

Simplify the right side.

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Step 4.6.2.3.3.1

Multiply $\frac{3}{4 + 2 \sqrt{3}}$ by $\frac{4 - 2 \sqrt{3}}{4 - 2 \sqrt{3}}$ .

$a = \frac{3}{4 + 2 \sqrt{3}} \cdot \frac{4 - 2 \sqrt{3}}{4 - 2 \sqrt{3}}$

Step 4.6.2.3.3.2

Multiply $\frac{3}{4 + 2 \sqrt{3}}$ by $\frac{4 - 2 \sqrt{3}}{4 - 2 \sqrt{3}}$ .

$a = \frac{3 (4 - 2 \sqrt{3})}{(4 + 2 \sqrt{3}) (4 - 2 \sqrt{3})}$

Step 4.6.2.3.3.3

Expand the denominator using the FOIL method.

$a = \frac{3 (4 - 2 \sqrt{3})}{16 - 8 \sqrt{3} + 8 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 4.6.2.3.3.4

Simplify.

$a = \frac{3 (4 - 2 \sqrt{3})}{4}$

Step 4.6.2.3.3.5

Cancel the common factor of $4 - 2 \sqrt{3}$ and $4$ .

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Step 4.6.2.3.3.5.1

Factor $2$ out of $3 (4 - 2 \sqrt{3})$ .

$a = \frac{2 (3 (2 - \sqrt{3}))}{4}$

Step 4.6.2.3.3.5.2

Cancel the common factors.

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Step 4.6.2.3.3.5.2.1

Factor $2$ out of $4$ .

$a = \frac{2 (3 (2 - \sqrt{3}))}{2 (2)}$

Step 4.6.2.3.3.5.2.2

Cancel the common factor.

$a = \frac{2 (3 (2 - \sqrt{3}))}{2 \cdot 2}$

Step 4.6.2.3.3.5.2.3

Rewrite the expression.

$a = \frac{3 (2 - \sqrt{3})}{2}$

Step 4.7

The final solution is all the values that make $(a - 3) (4 a + 2 a \sqrt{3} - 3) = 0$ true.

$a = 3, \frac{3 (2 - \sqrt{3})}{2}$

Step 5

Exclude the solutions that do not make $3 \cdot \sqrt{a} + a \cdot \sqrt{3} + a - 3 = 0$ true.

$a = \frac{3 (2 - \sqrt{3})}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3 (2 - \sqrt{3})}{2}$

Decimal Form:

$a = 0.40192378 \dots$