Basic Math Examples

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

Step 1

Subtract $2$ from $4$ .

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

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.

$1, a + 2$

Step 2.2

Remove parentheses.

$1, a + 2$

Step 2.3

The LCM of one and any expression is the expression.

$a + 2$

Step 3

Multiply each term in $f (a + 4) = \frac{3}{a + 2}$ by $a + 2$ to eliminate the fractions.

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Step 3.1

Multiply each term in $f (a + 4) = \frac{3}{a + 2}$ by $a + 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify by multiplying through.

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Move $4$ to the left of $f$ .

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

Step 3.2.2

Expand $(f a + 4 f) (a + 2)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.2.2.2

Apply the distributive property.

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

Step 3.2.2.3

Apply the distributive property.

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

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

Move $a$ .

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

Step 3.2.3.1.1.2

Multiply $a$ by $a$ .

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

Step 3.2.3.1.2

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

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

Step 3.2.3.1.3

Multiply $2$ by $4$ .

$f a^{2} + 2 f a + 4 f a + 8 f = \frac{3}{a + 2} (a + 2)$

Step 3.2.3.2

Add $2 f a$ and $4 f a$ .

$f a^{2} + 6 f a + 8 f = \frac{3}{a + 2} (a + 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $a + 2$ .

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Step 3.3.1.1

Cancel the common factor.

$f a^{2} + 6 f a + 8 f = \frac{3}{a + 2} (a + 2)$

Step 3.3.1.2

Rewrite the expression.

$f a^{2} + 6 f a + 8 f = 3$

Step 4

Solve the equation.

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Step 4.1

Subtract $3$ from both sides of the equation.

$f a^{2} + 6 f a + 8 f - 3 = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = f$ , $b = 6 f$ , and $c = 8 f - 3$ into the quadratic formula and solve for $a$ .

$\frac{- 6 f \pm \sqrt{{(6 f)}^{2} - 4 \cdot (f \cdot (8 f - 3))}}{2 f}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Add parentheses.

$a = \frac{- 6 f \pm \sqrt{{(6 f)}^{2} - 4 \cdot (f \cdot (8 f - 3))}}{2 f}$

Step 4.4.1.2

Let $u = f \cdot (8 f - 3)$ . Substitute $u$ for all occurrences of $f \cdot (8 f - 3)$ .

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Step 4.4.1.2.1

Apply the product rule to $6 f$ .

$a = \frac{- 6 f \pm \sqrt{6^{2} f^{2} - 4 \cdot u}}{2 f}$

Step 4.4.1.2.2

Raise $6$ to the power of $2$ .

$a = \frac{- 6 f \pm \sqrt{36 f^{2} - 4 u}}{2 f}$

Step 4.4.1.3

Factor $4$ out of $36 f^{2} - 4 u$ .

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Step 4.4.1.3.1

Factor $4$ out of $36 f^{2}$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2}) - 4 u}}{2 f}$

Step 4.4.1.3.2

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

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2}) + 4 (- u)}}{2 f}$

Step 4.4.1.3.3

Factor $4$ out of $4 (9 f^{2}) + 4 (- u)$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - u)}}{2 f}$

Step 4.4.1.4

Replace all occurrences of $u$ with $f \cdot (8 f - 3)$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (f \cdot (8 f - 3)))}}{2 f}$

Step 4.4.1.5

Simplify.

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Step 4.4.1.5.1

Simplify each term.

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Step 4.4.1.5.1.1

Apply the distributive property.

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (f (8 f) + f \cdot - 3))}}{2 f}$

Step 4.4.1.5.1.2

Rewrite using the commutative property of multiplication.

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 f \cdot f + f \cdot - 3))}}{2 f}$

Step 4.4.1.5.1.3

Move $- 3$ to the left of $f$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 f \cdot f - 3 \cdot f))}}{2 f}$

Step 4.4.1.5.1.4

Multiply $f$ by $f$ by adding the exponents.

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Step 4.4.1.5.1.4.1

Move $f$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 (f \cdot f) - 3 \cdot f))}}{2 f}$

Step 4.4.1.5.1.4.2

Multiply $f$ by $f$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 f^{2} - 3 \cdot f))}}{2 f}$

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 f^{2} - 3 f))}}{2 f}$

Step 4.4.1.5.1.5

Apply the distributive property.

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - (8 f^{2}) - (- 3 f))}}{2 f}$

Step 4.4.1.5.1.6

Multiply $8$ by $- 1$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - 8 f^{2} - (- 3 f))}}{2 f}$

Step 4.4.1.5.1.7

Multiply $- 3$ by $- 1$ .

$a = \frac{- 6 f \pm \sqrt{4 (9 f^{2} - 8 f^{2} + 3 f)}}{2 f}$

Step 4.4.1.5.2

Subtract $8 f^{2}$ from $9 f^{2}$ .

$a = \frac{- 6 f \pm \sqrt{4 (f^{2} + 3 f)}}{2 f}$

Step 4.4.1.6

Factor $f$ out of $f^{2} + 3 f$ .

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Step 4.4.1.6.1

Factor $f$ out of $f^{2}$ .

$a = \frac{- 6 f \pm \sqrt{4 (f \cdot f + 3 f)}}{2 f}$

Step 4.4.1.6.2

Factor $f$ out of $3 f$ .

$a = \frac{- 6 f \pm \sqrt{4 (f \cdot f + f \cdot 3)}}{2 f}$

Step 4.4.1.6.3

Factor $f$ out of $f \cdot f + f \cdot 3$ .

$a = \frac{- 6 f \pm \sqrt{4 (f (f + 3))}}{2 f}$

$a = \frac{- 6 f \pm \sqrt{4 f (f + 3)}}{2 f}$

Step 4.4.1.7

Rewrite $4 f (f + 3)$ as $2^{2} (f (f + 3))$ .

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Step 4.4.1.7.1

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

$a = \frac{- 6 f \pm \sqrt{2^{2} f (f + 3)}}{2 f}$

Step 4.4.1.7.2

Add parentheses.

$a = \frac{- 6 f \pm \sqrt{2^{2} (f (f + 3))}}{2 f}$

Step 4.4.1.8

Pull terms out from under the radical.

$a = \frac{- 6 f \pm 2 \sqrt{f (f + 3)}}{2 f}$

Step 4.4.2

Simplify $\frac{- 6 f \pm 2 \sqrt{f (f + 3)}}{2 f}$ .

$a = \frac{- 3 f \pm \sqrt{f (f + 3)}}{f}$

Step 4.5

The final answer is the combination of both solutions.

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

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

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

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