Basic Math Examples

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

Step 1

Subtract $2$ from both sides of the equation.

$a = \frac{3}{a + 2} - 2$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, a + 2, 1$

Step 2.2

Remove parentheses.

$1, a + 2, 1$

Step 2.3

The LCM of one and any expression is the expression.

$a + 2$

Step 3

Multiply each term in $a = \frac{3}{a + 2} - 2$ by $a + 2$ to eliminate the fractions.

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Step 3.1

Multiply each term in $a = \frac{3}{a + 2} - 2$ by $a + 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $a$ by $a$ .

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

Step 3.2.2.2

Move $2$ to the left of $a$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $a + 2$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

Step 3.3.1.1.2

Rewrite the expression.

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

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Multiply $- 2$ by $2$ .

$a^{2} + 2 a = 3 - 2 a - 4$

Step 3.3.2

Subtract $4$ from $3$ .

$a^{2} + 2 a = - 2 a - 1$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Add $2 a$ to both sides of the equation.

$a^{2} + 2 a + 2 a = - 1$

Step 4.1.2

Add $2 a$ and $2 a$ .

$a^{2} + 4 a = - 1$

Step 4.2

Add $1$ to both sides of the equation.

$a^{2} + 4 a + 1 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 1$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $a$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $4$ to the power of $2$ .

$a = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$a = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $1$ .

$a = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $4$ from $16$ .

$a = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.5.1.4.1

Factor $4$ out of $12$ .

$a = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$a = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$a = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

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

Step 4.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

$a = - 2 \pm \sqrt{3}$

Step 4.6

The final answer is the combination of both solutions.

$a = - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Decimal Form:

$a = - 0.26794919 \dots, - 3.73205080 \dots$