Trigonometry Examples

$(4 x^{2} - 8 x + 6) = (2 x - 1) \cdot (q (x)) + (\frac{3}{2 x - 1})$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(2 x - 1) \cdot (q (x)) + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2

Simplify $(2 x - 1) \cdot (q (x)) + \frac{3}{2 x - 1}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$2 x (q x) - 1 (q x) + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$2 (x \cdot x) q - 1 (q x) + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} q - 1 (q x) + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.1.3

Rewrite $- 1 (q x)$ as $- (q x)$ .

$2 x^{2} q - q x + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.2

To write $2 x^{2} q$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$- q x + \frac{2 x^{2} q (2 x - 1)}{2 x - 1} + \frac{3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.3

Combine the numerators over the common denominator.

$- q x + \frac{2 x^{2} q (2 x - 1) + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$- q x + \frac{2 x^{2} q (2 x) + 2 x^{2} q \cdot - 1 + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.2

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

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Step 2.4.2.1

Move $x$ .

$- q x + \frac{2 (x \cdot x^{2}) q \cdot 2 + 2 x^{2} q \cdot - 1 + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.2.2

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

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Step 2.4.2.2.1

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

$- q x + \frac{2 (x^{1} x^{2}) q \cdot 2 + 2 x^{2} q \cdot - 1 + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.2.2.2

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

$- q x + \frac{2 x^{1 + 2} q \cdot 2 + 2 x^{2} q \cdot - 1 + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.2.3

Add $1$ and $2$ .

$- q x + \frac{2 x^{3} q \cdot 2 + 2 x^{2} q \cdot - 1 + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.3

Multiply $- 1$ by $2$ .

$- q x + \frac{2 x^{3} q \cdot 2 - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.4.4

Multiply $2$ by $2$ .

$- q x + \frac{4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.5

To write $- q x$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$- q x \cdot \frac{2 x - 1}{2 x - 1} + \frac{4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.6

Simplify terms.

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Step 2.6.1

Combine $- q x$ and $\frac{2 x - 1}{2 x - 1}$ .

$\frac{- q x (2 x - 1)}{2 x - 1} + \frac{4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.6.2

Combine the numerators over the common denominator.

$\frac{- q x (2 x - 1) + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Apply the distributive property.

$\frac{- q x (2 x) - q x \cdot - 1 + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.2

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

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Step 2.7.2.1

Move $x$ .

$\frac{- q (x \cdot x) \cdot 2 - q x \cdot - 1 + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.2.2

Multiply $x$ by $x$ .

$\frac{- q x^{2} \cdot 2 - q x \cdot - 1 + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.3

Multiply $- q x \cdot - 1$ .

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Step 2.7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- q x^{2} \cdot 2 + 1 q x + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.3.2

Multiply $q$ by $1$ .

$\frac{- q x^{2} \cdot 2 + q x + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.4

Multiply $2$ by $- 1$ .

$\frac{- 2 q x^{2} + q x + 4 x^{3} q - 2 x^{2} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.5

Subtract $2 x^{2} q$ from $- 2 q x^{2}$ .

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Step 2.7.5.1

Move $x^{2}$ .

$\frac{- 2 q x^{2} - 2 q x^{2} + q x + 4 x^{3} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.7.5.2

Subtract $2 q x^{2}$ from $- 2 q x^{2}$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8

Simplify with factoring out.

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Step 2.8.1

Factor $- 1$ out of $- 4 q x^{2}$ .

$\frac{- (4 q x^{2}) + q x + 4 x^{3} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.2

Factor $- 1$ out of $q x$ .

$\frac{- (4 q x^{2}) - (- q x) + 4 x^{3} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.3

Factor $- 1$ out of $- (4 q x^{2}) - (- q x)$ .

$\frac{- (4 q x^{2} - q x) + 4 x^{3} q + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.4

Factor $- 1$ out of $4 x^{3} q$ .

$\frac{- (4 q x^{2} - q x) - (- 4 x^{3} q) + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.5

Factor $- 1$ out of $- (4 q x^{2} - q x) - (- 4 x^{3} q)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q) + 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.6

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q) - 1 (- 3)}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.7

Factor $- 1$ out of $- (4 q x^{2} - q x - 4 x^{3} q) - 1 (- 3)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3)}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.8

Simplify the expression.

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Step 2.8.8.1

Rewrite $- (4 q x^{2} - q x - 4 x^{3} q - 3)$ as $- 1 (4 q x^{2} - q x - 4 x^{3} q - 3)$ .

$\frac{- 1 (4 q x^{2} - q x - 4 x^{3} q - 3)}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 2.8.8.2

Move the negative in front of the fraction.

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3}{2 x - 1} = 4 x^{2} - 8 x + 6$

Step 3

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

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Step 3.1

Subtract $4 x^{2}$ from both sides of the equation.

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3}{2 x - 1} - 4 x^{2} = - 8 x + 6$

Step 3.2

Add $8 x$ to both sides of the equation.

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3}{2 x - 1} - 4 x^{2} + 8 x = 6$

Step 3.3

To write $- 4 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3}{2 x - 1} - 4 x^{2} \cdot \frac{2 x - 1}{2 x - 1} + 8 x = 6$

Step 3.4

Combine $- 4 x^{2}$ and $\frac{2 x - 1}{2 x - 1}$ .

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3}{2 x - 1} + \frac{- 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3) - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{- (4 q x^{2}) - (- q x) - (- 4 x^{3} q) - - 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.2

Simplify.

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Step 3.6.2.1

Multiply $4$ by $- 1$ .

$\frac{- 4 (q x^{2}) - (- q x) - (- 4 x^{3} q) - - 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.2.2

Multiply $- (- q x)$ .

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Step 3.6.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- 4 (q x^{2}) + 1 (q x) - (- 4 x^{3} q) - - 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.2.2.2

Multiply $q$ by $1$ .

$\frac{- 4 (q x^{2}) + q x - (- 4 x^{3} q) - - 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.2.3

Multiply $- 4$ by $- 1$ .

$\frac{- 4 (q x^{2}) + q x + 4 (x^{3} q) - - 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.2.4

Multiply $- 1$ by $- 3$ .

$\frac{- 4 (q x^{2}) + q x + 4 (x^{3} q) + 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.3

Remove parentheses.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 x^{2} (2 x - 1)}{2 x - 1} + 8 x = 6$

Step 3.6.4

Apply the distributive property.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 x^{2} (2 x) - 4 x^{2} \cdot - 1}{2 x - 1} + 8 x = 6$

Step 3.6.5

Rewrite using the commutative property of multiplication.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 x^{2} x - 4 x^{2} \cdot - 1}{2 x - 1} + 8 x = 6$

Step 3.6.6

Multiply $- 1$ by $- 4$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 x^{2} x + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.6.7

Simplify each term.

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Step 3.6.7.1

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

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Step 3.6.7.1.1

Move $x$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 (x \cdot x^{2}) + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.6.7.1.2

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

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Step 3.6.7.1.2.1

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

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 (x^{1} x^{2}) + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.6.7.1.2.2

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

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 x^{1 + 2} + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.6.7.1.3

Add $1$ and $2$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 4 \cdot 2 x^{3} + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.6.7.2

Multiply $- 4$ by $2$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2}}{2 x - 1} + 8 x = 6$

Step 3.7

To write $8 x$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2}}{2 x - 1} + \frac{8 x (2 x - 1)}{2 x - 1} = 6$

Step 3.8

Combine the numerators over the common denominator.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 x (2 x - 1)}{2 x - 1} = 6$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Apply the distributive property.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 x (2 x) + 8 x \cdot - 1}{2 x - 1} = 6$

Step 3.9.2

Rewrite using the commutative property of multiplication.

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 \cdot 2 x \cdot x + 8 x \cdot - 1}{2 x - 1} = 6$

Step 3.9.3

Multiply $- 1$ by $8$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 \cdot 2 x \cdot x - 8 x}{2 x - 1} = 6$

Step 3.9.4

Simplify each term.

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Step 3.9.4.1

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

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Step 3.9.4.1.1

Move $x$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 \cdot 2 (x \cdot x) - 8 x}{2 x - 1} = 6$

Step 3.9.4.1.2

Multiply $x$ by $x$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 8 \cdot 2 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.9.4.2

Multiply $8$ by $2$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 4 x^{2} + 16 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.9.5

Add $4 x^{2}$ and $16 x^{2}$ .

$\frac{- 4 q x^{2} + q x + 4 x^{3} q + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.10

Factor $- 1$ out of $- 4 q x^{2}$ .

$\frac{- (4 q x^{2}) + q x + 4 x^{3} q + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.11

Factor $- 1$ out of $q x$ .

$\frac{- (4 q x^{2}) - (- q x) + 4 x^{3} q + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.12

Factor $- 1$ out of $- (4 q x^{2}) - (- q x)$ .

$\frac{- (4 q x^{2} - q x) + 4 x^{3} q + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.13

Factor $- 1$ out of $4 x^{3} q$ .

$\frac{- (4 q x^{2} - q x) - (- 4 x^{3} q) + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.14

Factor $- 1$ out of $- (4 q x^{2} - q x) - (- 4 x^{3} q)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q) + 3 - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.15

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q) - 1 (- 3) - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.16

Factor $- 1$ out of $- (4 q x^{2} - q x - 4 x^{3} q) - 1 (- 3)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3) - 8 x^{3} + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.17

Factor $- 1$ out of $- 8 x^{3}$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3) - (8 x^{3}) + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.18

Factor $- 1$ out of $- (4 q x^{2} - q x - 4 x^{3} q - 3) - (8 x^{3})$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3}) + 20 x^{2} - 8 x}{2 x - 1} = 6$

Step 3.19

Factor $- 1$ out of $20 x^{2}$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3}) - (- 20 x^{2}) - 8 x}{2 x - 1} = 6$

Step 3.20

Factor $- 1$ out of $- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3}) - (- 20 x^{2})$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2}) - 8 x}{2 x - 1} = 6$

Step 3.21

Factor $- 1$ out of $- 8 x$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2}) - (8 x)}{2 x - 1} = 6$

Step 3.22

Factor $- 1$ out of $- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2}) - (8 x)$ .

$\frac{- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2} + 8 x)}{2 x - 1} = 6$

Step 3.23

Rewrite $- (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2} + 8 x)$ as $- 1 (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2} + 8 x)$ .

$\frac{- 1 (4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2} + 8 x)}{2 x - 1} = 6$

Step 3.24

Move the negative in front of the fraction.

$- \frac{4 q x^{2} - q x - 4 x^{3} q - 3 + 8 x^{3} - 20 x^{2} + 8 x}{2 x - 1} = 6$