Finite Math Examples

$\frac{f}{g} \cdot (x) = \frac{5 x^{2} - 3 x + 4}{2 x^{2} - 7 x}$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $\frac{5 x^{2} - 3 x + 4}{2 x^{2} - 7 x}$ from both sides of the equation.

$\frac{f}{g} \cdot x - \frac{5 x^{2} - 3 x + 4}{2 x^{2} - 7 x} = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Combine $\frac{f}{g}$ and $x$ .

$\frac{f x}{g} - \frac{5 x^{2} - 3 x + 4}{2 x^{2} - 7 x} = 0$

Step 1.2.2

Factor $x$ out of $2 x^{2} - 7 x$ .

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Step 1.2.2.1

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

$\frac{f x}{g} - \frac{5 x^{2} - 3 x + 4}{x (2 x) - 7 x} = 0$

Step 1.2.2.2

Factor $x$ out of $- 7 x$ .

$\frac{f x}{g} - \frac{5 x^{2} - 3 x + 4}{x (2 x) + x \cdot - 7} = 0$

Step 1.2.2.3

Factor $x$ out of $x (2 x) + x \cdot - 7$ .

$\frac{f x}{g} - \frac{5 x^{2} - 3 x + 4}{x (2 x - 7)} = 0$

Step 1.3

To write $\frac{f x}{g}$ as a fraction with a common denominator, multiply by $\frac{x (2 x - 7)}{x (2 x - 7)}$ .

$\frac{f x}{g} \cdot \frac{x (2 x - 7)}{x (2 x - 7)} - \frac{5 x^{2} - 3 x + 4}{x (2 x - 7)} = 0$

Step 1.4

To write $- \frac{5 x^{2} - 3 x + 4}{x (2 x - 7)}$ as a fraction with a common denominator, multiply by $\frac{g}{g}$ .

$\frac{f x}{g} \cdot \frac{x (2 x - 7)}{x (2 x - 7)} - \frac{5 x^{2} - 3 x + 4}{x (2 x - 7)} \cdot \frac{g}{g} = 0$

Step 1.5

Write each expression with a common denominator of $g x (2 x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{f x}{g}$ by $\frac{x (2 x - 7)}{x (2 x - 7)}$ .

$\frac{f x (x (2 x - 7))}{g (x (2 x - 7))} - \frac{5 x^{2} - 3 x + 4}{x (2 x - 7)} \cdot \frac{g}{g} = 0$

Step 1.5.2

Multiply $\frac{5 x^{2} - 3 x + 4}{x (2 x - 7)}$ by $\frac{g}{g}$ .

$\frac{f x (x (2 x - 7))}{g (x (2 x - 7))} - \frac{(5 x^{2} - 3 x + 4) g}{x (2 x - 7) g} = 0$

Step 1.5.3

Reorder the factors of $g (x (2 x - 7))$ .

$\frac{f x (x (2 x - 7))}{g x (2 x - 7)} - \frac{(5 x^{2} - 3 x + 4) g}{x (2 x - 7) g} = 0$

Step 1.5.4

Reorder the factors of $x (2 x - 7) g$ .

$\frac{f x (x (2 x - 7))}{g x (2 x - 7)} - \frac{(5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.6

Combine the numerators over the common denominator.

$\frac{f x (x (2 x - 7)) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7

Simplify the numerator.

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Step 1.7.1

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

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Step 1.7.1.1

Move $x$ .

$\frac{f (x \cdot x) (2 x - 7) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.1.2

Multiply $x$ by $x$ .

$\frac{f x^{2} (2 x - 7) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.2

Apply the distributive property.

$\frac{f x^{2} (2 x) + f x^{2} \cdot - 7 - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.3

Rewrite using the commutative property of multiplication.

$\frac{2 f x^{2} x + f x^{2} \cdot - 7 - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.4

Move $- 7$ to the left of $f x^{2}$ .

$\frac{2 f x^{2} x - 7 \cdot (f x^{2}) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.5

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

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Step 1.7.5.1

Move $x$ .

$\frac{2 f (x \cdot x^{2}) - 7 \cdot (f x^{2}) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.5.2

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

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Step 1.7.5.2.1

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

$\frac{2 f (x^{1} x^{2}) - 7 \cdot (f x^{2}) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.5.2.2

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

$\frac{2 f x^{1 + 2} - 7 \cdot (f x^{2}) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.5.3

Add $1$ and $2$ .

$\frac{2 f x^{3} - 7 \cdot (f x^{2}) - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

$\frac{2 f x^{3} - 7 f x^{2} - (5 x^{2} - 3 x + 4) g}{g x (2 x - 7)} = 0$

Step 1.7.6

Apply the distributive property.

$\frac{2 f x^{3} - 7 f x^{2} + (- (5 x^{2}) - (- 3 x) - 1 \cdot 4) g}{g x (2 x - 7)} = 0$

Step 1.7.7

Simplify.

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Step 1.7.7.1

Multiply $5$ by $- 1$ .

$\frac{2 f x^{3} - 7 f x^{2} + (- 5 x^{2} - (- 3 x) - 1 \cdot 4) g}{g x (2 x - 7)} = 0$

Step 1.7.7.2

Multiply $- 3$ by $- 1$ .

$\frac{2 f x^{3} - 7 f x^{2} + (- 5 x^{2} + 3 x - 1 \cdot 4) g}{g x (2 x - 7)} = 0$

Step 1.7.7.3

Multiply $- 1$ by $4$ .

$\frac{2 f x^{3} - 7 f x^{2} + (- 5 x^{2} + 3 x - 4) g}{g x (2 x - 7)} = 0$

Step 1.7.8

Apply the distributive property.

$\frac{2 f x^{3} - 7 f x^{2} - 5 x^{2} g + 3 x g - 4 g}{g x (2 x - 7)} = 0$

Step 2

Set the numerator equal to zero.

$2 f x^{3} - 7 f x^{2} - 5 x^{2} g + 3 x g - 4 g = 0$

Step 3

Solve the equation for $f$ .

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Step 3.1

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

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Step 3.1.1

Add $5 x^{2} g$ to both sides of the equation.

$2 f x^{3} - 7 f x^{2} + 3 x g - 4 g = 5 x^{2} g$

Step 3.1.2

Subtract $3 x g$ from both sides of the equation.

$2 f x^{3} - 7 f x^{2} - 4 g = 5 x^{2} g - 3 x g$

Step 3.1.3

Add $4 g$ to both sides of the equation.

$2 f x^{3} - 7 f x^{2} = 5 x^{2} g - 3 x g + 4 g$

Step 3.2

Factor $f x^{2}$ out of $2 f x^{3} - 7 f x^{2}$ .

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Step 3.2.1

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

$f x^{2} (2 x) - 7 f x^{2} = 5 x^{2} g - 3 x g + 4 g$

Step 3.2.2

Factor $f x^{2}$ out of $- 7 f x^{2}$ .

$f x^{2} (2 x) + f x^{2} (- 7) = 5 x^{2} g - 3 x g + 4 g$

Step 3.2.3

Factor $f x^{2}$ out of $f x^{2} (2 x) + f x^{2} (- 7)$ .

$f x^{2} (2 x - 7) = 5 x^{2} g - 3 x g + 4 g$

Step 3.3

Divide each term in $f x^{2} (2 x - 7) = 5 x^{2} g - 3 x g + 4 g$ by $x^{2} (2 x - 7)$ and simplify.

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Step 3.3.1

Divide each term in $f x^{2} (2 x - 7) = 5 x^{2} g - 3 x g + 4 g$ by $x^{2} (2 x - 7)$ .

$\frac{f x^{2} (2 x - 7)}{x^{2} (2 x - 7)} = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{f x^{2} (2 x - 7)}{x^{2} (2 x - 7)} = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.2.1.2

Rewrite the expression.

$\frac{f (2 x - 7)}{2 x - 7} = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.2.2

Cancel the common factor of $2 x - 7$ .

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Step 3.3.2.2.1

Cancel the common factor.

$\frac{f (2 x - 7)}{2 x - 7} = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.2.2.2

Divide $f$ by $1$ .

$f = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

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Step 3.3.3.1.1.1

Cancel the common factor.

$f = \frac{5 x^{2} g}{x^{2} (2 x - 7)} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.1.2

Rewrite the expression.

$f = \frac{5 g}{2 x - 7} + \frac{- 3 x g}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.2

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

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Step 3.3.3.1.2.1

Factor $x$ out of $- 3 x g$ .

$f = \frac{5 g}{2 x - 7} + \frac{x (- 3 g)}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.1.2.2.1

Factor $x$ out of $x^{2} (2 x - 7)$ .

$f = \frac{5 g}{2 x - 7} + \frac{x (- 3 g)}{x (x (2 x - 7))} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.2.2.2

Cancel the common factor.

$f = \frac{5 g}{2 x - 7} + \frac{x (- 3 g)}{x (x (2 x - 7))} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.2.2.3

Rewrite the expression.

$f = \frac{5 g}{2 x - 7} + \frac{- 3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.1.3

Move the negative in front of the fraction.

$f = \frac{5 g}{2 x - 7} - \frac{3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.2

To write $\frac{5 g}{2 x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f = \frac{5 g}{2 x - 7} \cdot \frac{x}{x} - \frac{3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.3

Write each expression with a common denominator of $(2 x - 7) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.3.1

Multiply $\frac{5 g}{2 x - 7}$ by $\frac{x}{x}$ .

$f = \frac{5 g x}{(2 x - 7) x} - \frac{3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.3.2

Reorder the factors of $(2 x - 7) x$ .

$f = \frac{5 g x}{x (2 x - 7)} - \frac{3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.4

Combine the numerators over the common denominator.

$f = \frac{5 g x - 3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.5

Factor $g$ out of $5 g x - 3 g$ .

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Step 3.3.3.5.1

Factor $g$ out of $5 g x$ .

$f = \frac{g (5 x) - 3 g}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.5.2

Factor $g$ out of $- 3 g$ .

$f = \frac{g (5 x) + g \cdot - 3}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.5.3

Factor $g$ out of $g (5 x) + g \cdot - 3$ .

$f = \frac{g (5 x - 3)}{x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.6

To write $\frac{g (5 x - 3)}{x (2 x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f = \frac{g (5 x - 3)}{x (2 x - 7)} \cdot \frac{x}{x} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.7

Write each expression with a common denominator of $(2 x - 7) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.7.1

Multiply $\frac{g (5 x - 3)}{x (2 x - 7)}$ by $\frac{x}{x}$ .

$f = \frac{g (5 x - 3) x}{x (2 x - 7) x} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.7.2

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

$f = \frac{g (5 x - 3) x}{x^{1} x (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.7.3

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

$f = \frac{g (5 x - 3) x}{x^{1} x^{1} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.7.4

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

$f = \frac{g (5 x - 3) x}{x^{1 + 1} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.7.5

Add $1$ and $1$ .

$f = \frac{g (5 x - 3) x}{x^{2} (2 x - 7)} + \frac{4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.8

Combine the numerators over the common denominator.

$f = \frac{g (5 x - 3) x + 4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.9

Simplify the numerator.

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Step 3.3.3.9.1

Factor $g$ out of $g (5 x - 3) x + 4 g$ .

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Step 3.3.3.9.1.1

Factor $g$ out of $g (5 x - 3) x$ .

$f = \frac{g ((5 x - 3) x) + 4 g}{x^{2} (2 x - 7)}$

Step 3.3.3.9.1.2

Factor $g$ out of $4 g$ .

$f = \frac{g ((5 x - 3) x) + g \cdot 4}{x^{2} (2 x - 7)}$

Step 3.3.3.9.1.3

Factor $g$ out of $g ((5 x - 3) x) + g \cdot 4$ .

$f = \frac{g ((5 x - 3) x + 4)}{x^{2} (2 x - 7)}$

Step 3.3.3.9.2

Apply the distributive property.

$f = \frac{g (5 x \cdot x - 3 x + 4)}{x^{2} (2 x - 7)}$

Step 3.3.3.9.3

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

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Step 3.3.3.9.3.1

Move $x$ .

$f = \frac{g (5 (x \cdot x) - 3 x + 4)}{x^{2} (2 x - 7)}$

Step 3.3.3.9.3.2

Multiply $x$ by $x$ .

$f = \frac{g (5 x^{2} - 3 x + 4)}{x^{2} (2 x - 7)}$