Algebra Examples

$f^{- 1} (\frac{3 + x}{x - 2}) = x + 1$

Step 1

Simplify $f^{- 1} (\frac{3 + x}{x - 2})$ .

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{f} \cdot \frac{3 + x}{x - 2} = x + 1$

Step 1.2

Multiply $\frac{1}{f}$ by $\frac{3 + x}{x - 2}$ .

$\frac{3 + x}{f (x - 2)} = x + 1$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$\frac{3 + x}{f (x - 2)} - x = 1$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{f (x - 2)}{f (x - 2)}$ .

$\frac{3 + x}{f (x - 2)} - x \cdot \frac{f (x - 2)}{f (x - 2)} = 1$

Step 2.3

Combine $- x$ and $\frac{f (x - 2)}{f (x - 2)}$ .

$\frac{3 + x}{f (x - 2)} + \frac{- x (f (x - 2))}{f (x - 2)} = 1$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 + x - x (f (x - 2))}{f (x - 2)} = 1$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{3 + x - x (f x + f \cdot - 2)}{f (x - 2)} = 1$

Step 2.5.2

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

$\frac{3 + x - x (f x - 2 \cdot f)}{f (x - 2)} = 1$

Step 2.5.3

Apply the distributive property.

$\frac{3 + x - x (f x) - x (- 2 f)}{f (x - 2)} = 1$

Step 2.5.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.4.1

Move $x$ .

$\frac{3 + x - (x \cdot x) f - x (- 2 f)}{f (x - 2)} = 1$

Step 2.5.4.2

Multiply $x$ by $x$ .

$\frac{3 + x - x^{2} f - x (- 2 f)}{f (x - 2)} = 1$

Step 2.5.5

Rewrite using the commutative property of multiplication.

$\frac{3 + x - x^{2} f - 1 \cdot - 2 x f}{f (x - 2)} = 1$

Step 2.5.6

Multiply $- 1$ by $- 2$ .

$\frac{3 + x - x^{2} f + 2 x f}{f (x - 2)} = 1$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$f (x - 2), 1$

Step 3.2

The LCM of one and any expression is the expression.

$f (x - 2)$

Step 4

Multiply each term in $\frac{3 + x - x^{2} f + 2 x f}{f (x - 2)} = 1$ by $f (x - 2)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{3 + x - x^{2} f + 2 x f}{f (x - 2)} = 1$ by $f (x - 2)$ .

$\frac{3 + x - x^{2} f + 2 x f}{f (x - 2)} (f (x - 2)) = 1 (f (x - 2))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $f (x - 2)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 + x - x^{2} f + 2 x f}{f (x - 2)} (f (x - 2)) = 1 (f (x - 2))$

Step 4.2.1.2

Rewrite the expression.

$3 + x - x^{2} f + 2 x f = 1 (f (x - 2))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $f (x - 2)$ by $1$ .

$3 + x - x^{2} f + 2 x f = f (x - 2)$

Step 4.3.2

Apply the distributive property.

$3 + x - x^{2} f + 2 x f = f x + f \cdot - 2$

Step 4.3.3

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

$3 + x - x^{2} f + 2 x f = f x - 2 f$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

Subtract $f x$ from both sides of the equation.

$3 + x - x^{2} f + 2 x f - f x = - 2 f$

Step 5.1.2

Subtract $f x$ from $2 x f$ .

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Step 5.1.2.1

Move $x$ .

$3 + x - x^{2} f + 2 f x - f x = - 2 f$

Step 5.1.2.2

Subtract $f x$ from $2 f x$ .

$3 + x - x^{2} f + 1 f x = - 2 f$

Step 5.1.3

Multiply $f$ by $1$ .

$3 + x - x^{2} f + f x = - 2 f$

Step 5.2

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

$3 + x - x^{2} f + f x + 2 f = 0$

Step 5.3

Use the quadratic formula to find the solutions.

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

Step 5.4

Substitute the values $a = - f$ , $b = 1 + f$ , and $c = 3 + 2 f$ into the quadratic formula and solve for $x$ .

$\frac{- (1 + f) \pm \sqrt{{(1 + f)}^{2} - 4 \cdot (- f \cdot (3 + 2 f))}}{2 (- f)}$

Step 5.5

Simplify.

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Apply the distributive property.

$x = \frac{- 1 \cdot 1 - f \pm \sqrt{{(1 + f)}^{2} - 4 \cdot (- 1 f) \cdot (3 + 2 f)}}{2 (- f)}$

Step 5.5.1.2

Multiply $- 1$ by $1$ .

$x = \frac{- 1 - f \pm \sqrt{{(1 + f)}^{2} - 4 \cdot (- 1 f) \cdot (3 + 2 f)}}{2 (- f)}$

Step 5.5.1.3

Rewrite ${(1 + f)}^{2}$ as $(1 + f) (1 + f)$ .

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

Step 5.5.1.4

Expand $(1 + f) (1 + f)$ using the FOIL Method.

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Step 5.5.1.4.1

Apply the distributive property.

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

Step 5.5.1.4.2

Apply the distributive property.

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

Step 5.5.1.4.3

Apply the distributive property.

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

Step 5.5.1.5

Simplify and combine like terms.

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Step 5.5.1.5.1

Simplify each term.

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Step 5.5.1.5.1.1

Multiply $1$ by $1$ .

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

Step 5.5.1.5.1.2

Multiply $f$ by $1$ .

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

Step 5.5.1.5.1.3

Multiply $f$ by $1$ .

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

Step 5.5.1.5.1.4

Multiply $f$ by $f$ .

$x = \frac{- 1 - f \pm \sqrt{1 + f + f + f^{2} - 4 \cdot (- 1 f) \cdot (3 + 2 f)}}{2 (- f)}$

Step 5.5.1.5.2

Add $f$ and $f$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} - 4 \cdot (- 1 f) \cdot (3 + 2 f)}}{2 (- f)}$

Step 5.5.1.6

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 4 f \cdot (3 + 2 f)}}{2 (- f)}$

Step 5.5.1.7

Apply the distributive property.

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 4 f \cdot 3 + 4 f (2 f)}}{2 (- f)}$

Step 5.5.1.8

Multiply $3$ by $4$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 12 f + 4 f (2 f)}}{2 (- f)}$

Step 5.5.1.9

Rewrite using the commutative property of multiplication.

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 12 f + 4 \cdot (2 f \cdot f)}}{2 (- f)}$

Step 5.5.1.10

Simplify each term.

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Step 5.5.1.10.1

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

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Step 5.5.1.10.1.1

Move $f$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 12 f + 4 \cdot (2 (f \cdot f))}}{2 (- f)}$

Step 5.5.1.10.1.2

Multiply $f$ by $f$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 12 f + 4 \cdot (2 f^{2})}}{2 (- f)}$

Step 5.5.1.10.2

Multiply $4$ by $2$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 2 f + f^{2} + 12 f + 8 f^{2}}}{2 (- f)}$

Step 5.5.1.11

Add $2 f$ and $12 f$ .

$x = \frac{- 1 - f \pm \sqrt{1 + f^{2} + 14 f + 8 f^{2}}}{2 (- f)}$

Step 5.5.1.12

Add $f^{2}$ and $8 f^{2}$ .

$x = \frac{- 1 - f \pm \sqrt{1 + 9 f^{2} + 14 f}}{2 (- f)}$

Step 5.5.1.13

Reorder terms.

$x = \frac{- 1 - f \pm \sqrt{9 f^{2} + 14 f + 1}}{2 (- f)}$

Step 5.5.2

Multiply $- 1$ by $2$ .

$x = \frac{- 1 - f \pm \sqrt{9 f^{2} + 14 f + 1}}{- 2 f}$

Step 5.5.3

Simplify $\frac{- 1 - f \pm \sqrt{9 f^{2} + 14 f + 1}}{- 2 f}$ .

$x = \frac{1 + f \pm \sqrt{9 f^{2} + 14 f + 1}}{2 f}$

Step 5.6

The final answer is the combination of both solutions.

$x = \frac{1 + f + \sqrt{9 f^{2} + 14 f + 1}}{2 f}$

$x = \frac{1 + f - \sqrt{9 f^{2} + 14 f + 1}}{2 f}$

$x = \frac{1 + f + \sqrt{9 f^{2} + 14 f + 1}}{2 f}$

$x = \frac{1 + f - \sqrt{9 f^{2} + 14 f + 1}}{2 f}$