Calculus Examples

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

Step 1

Rewrite the equation as $\frac{x^{2} - 4 x + 3}{x^{2} + a x + b} = f (x)$ .

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

Step 2

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.2

Write the factored form using these integers.

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

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.

$x^{2} + a x + b, 1$

Step 3.2

Remove parentheses.

$x^{2} + a x + b, 1$

Step 3.3

The LCM of one and any expression is the expression.

$x^{2} + a x + b$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $x^{2} + a x + b$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Rewrite the expression.

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

Step 4.2.2

Expand $(x - 3) (x - 1)$ using the FOIL Method.

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Step 4.2.2.1

Apply the distributive property.

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

Step 4.2.2.2

Apply the distributive property.

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

Step 4.2.2.3

Apply the distributive property.

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

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $x$ by $x$ .

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

Step 4.2.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 4.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.2.3.1.4

Multiply $- 3$ by $- 1$ .

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

Step 4.2.3.2

Subtract $3 x$ from $- x$ .

$x^{2} - 4 x + 3 = f (x) (x^{2} + a x + b)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$x^{2} - 4 x + 3 = f (x) x^{2} + f (x) (a x) + f (x) b$

Step 4.3.2

Reorder factors in $f (x) x^{2} + f (x) a x + f (x) b$ .

$x^{2} - 4 x + 3 = x^{2} f (x) + a x f (x) + b f (x)$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $x^{2} f (x) + a x f (x) + b f x = x^{2} - 4 x + 3$ .

$x^{2} f (x) + a x f (x) + b f x = x^{2} - 4 x + 3$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.2.1.1

Move $x$ .

$x \cdot x^{2} f + a x f (x) + b f x = x^{2} - 4 x + 3$

Step 5.2.1.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.1.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{2} f + a x f (x) + b f x = x^{2} - 4 x + 3$

Step 5.2.1.2.2

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

$x^{1 + 2} f + a x f (x) + b f x = x^{2} - 4 x + 3$

Step 5.2.1.3

Add $1$ and $2$ .

$x^{3} f + a x f (x) + b f x = x^{2} - 4 x + 3$

Step 5.2.2

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

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Step 5.2.2.1

Move $x$ .

$x^{3} f + a (x \cdot x) f + b f x = x^{2} - 4 x + 3$

Step 5.2.2.2

Multiply $x$ by $x$ .

$x^{3} f + a x^{2} f + b f x = x^{2} - 4 x + 3$

Step 5.3

Move all terms not containing $b$ to the right side of the equation.

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Step 5.3.1

Subtract $x^{3} f$ from both sides of the equation.

$a x^{2} f + b f x = x^{2} - 4 x + 3 - x^{3} f$

Step 5.3.2

Subtract $a x^{2} f$ from both sides of the equation.

$b f x = x^{2} - 4 x + 3 - x^{3} f - a x^{2} f$

Step 5.4

Divide each term in $b f x = x^{2} - 4 x + 3 - x^{3} f - a x^{2} f$ by $f x$ and simplify.

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Step 5.4.1

Divide each term in $b f x = x^{2} - 4 x + 3 - x^{3} f - a x^{2} f$ by $f x$ .

$\frac{b f x}{f x} = \frac{x^{2}}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $f$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{b f x}{f x} = \frac{x^{2}}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{b x}{x} = \frac{x^{2}}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.2.2

Cancel the common factor of $x$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{b x}{x} = \frac{x^{2}}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.2.2.2

Divide $b$ by $1$ .

$b = \frac{x^{2}}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.4.3.1.1.1

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

$b = \frac{x \cdot x}{f x} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $x$ out of $f x$ .

$b = \frac{x \cdot x}{x f} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$b = \frac{x \cdot x}{x f} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$b = \frac{x}{f} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.2

Cancel the common factor of $x$ .

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Step 5.4.3.1.2.1

Cancel the common factor.

$b = \frac{x}{f} + \frac{- 4 x}{f x} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.2.2

Rewrite the expression.

$b = \frac{x}{f} + \frac{- 4}{f} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.3

Move the negative in front of the fraction.

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} + \frac{- x^{3} f}{f x} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.4

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.4.3.1.4.1

Factor $x$ out of $- x^{3} f$ .

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

Step 5.4.3.1.4.2

Cancel the common factors.

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Step 5.4.3.1.4.2.1

Factor $x$ out of $f x$ .

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

Step 5.4.3.1.4.2.2

Cancel the common factor.

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

Step 5.4.3.1.4.2.3

Rewrite the expression.

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} + \frac{- x^{2} f}{f} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.5

Cancel the common factor of $f$ .

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Step 5.4.3.1.5.1

Cancel the common factor.

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} + \frac{- x^{2} f}{f} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.5.2

Divide $- x^{2}$ by $1$ .

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} - x^{2} + \frac{- a x^{2} f}{f x}$

Step 5.4.3.1.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.4.3.1.6.1

Factor $x$ out of $- a x^{2} f$ .

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

Step 5.4.3.1.6.2

Cancel the common factors.

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Step 5.4.3.1.6.2.1

Factor $x$ out of $f x$ .

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

Step 5.4.3.1.6.2.2

Cancel the common factor.

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

Step 5.4.3.1.6.2.3

Rewrite the expression.

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} - x^{2} + \frac{- a x f}{f}$

Step 5.4.3.1.7

Cancel the common factor of $f$ .

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Step 5.4.3.1.7.1

Cancel the common factor.

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} - x^{2} + \frac{- a x f}{f}$

Step 5.4.3.1.7.2

Divide $- a x$ by $1$ .

$b = \frac{x}{f} - \frac{4}{f} + \frac{3}{f x} - x^{2} - a x$