Trigonometry Examples

$- 2 x (3 x + 11) (2 x - 5) = 0$

Step 1

Simplify $- 2 x (3 x + 11) (2 x - 5)$ .

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Step 1.1

Simplify terms.

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Step 1.1.1

Apply the distributive property.

$(- 2 x (3 x) - 2 x \cdot 11) (2 x - 5) = 0$

Step 1.1.2

Simplify the expression.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$(- 2 \cdot 3 x \cdot x - 2 x \cdot 11) (2 x - 5) = 0$

Step 1.1.2.2

Multiply $11$ by $- 2$ .

$(- 2 \cdot 3 x \cdot x - 22 x) (2 x - 5) = 0$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

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

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Step 1.1.3.1.1

Move $x$ .

$(- 2 \cdot 3 (x \cdot x) - 22 x) (2 x - 5) = 0$

Step 1.1.3.1.2

Multiply $x$ by $x$ .

$(- 2 \cdot 3 x^{2} - 22 x) (2 x - 5) = 0$

Step 1.1.3.2

Multiply $- 2$ by $3$ .

$(- 6 x^{2} - 22 x) (2 x - 5) = 0$

Step 1.2

Expand $(- 6 x^{2} - 22 x) (2 x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$- 6 x^{2} (2 x - 5) - 22 x (2 x - 5) = 0$

Step 1.2.2

Apply the distributive property.

$- 6 x^{2} (2 x) - 6 x^{2} \cdot - 5 - 22 x (2 x - 5) = 0$

Step 1.2.3

Apply the distributive property.

$- 6 x^{2} (2 x) - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$- 6 \cdot 2 x^{2} x - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

$- 6 \cdot 2 (x \cdot x^{2}) - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.2.2

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

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Step 1.3.1.2.2.1

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

$- 6 \cdot 2 (x^{1} x^{2}) - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.2.2.2

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

$- 6 \cdot 2 x^{1 + 2} - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.2.3

Add $1$ and $2$ .

$- 6 \cdot 2 x^{3} - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.3

Multiply $- 6$ by $2$ .

$- 12 x^{3} - 6 x^{2} \cdot - 5 - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.4

Multiply $- 5$ by $- 6$ .

$- 12 x^{3} + 30 x^{2} - 22 x (2 x) - 22 x \cdot - 5 = 0$

Step 1.3.1.5

Rewrite using the commutative property of multiplication.

$- 12 x^{3} + 30 x^{2} - 22 \cdot 2 x \cdot x - 22 x \cdot - 5 = 0$

Step 1.3.1.6

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

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Step 1.3.1.6.1

Move $x$ .

$- 12 x^{3} + 30 x^{2} - 22 \cdot 2 (x \cdot x) - 22 x \cdot - 5 = 0$

Step 1.3.1.6.2

Multiply $x$ by $x$ .

$- 12 x^{3} + 30 x^{2} - 22 \cdot 2 x^{2} - 22 x \cdot - 5 = 0$

Step 1.3.1.7

Multiply $- 22$ by $2$ .

$- 12 x^{3} + 30 x^{2} - 44 x^{2} - 22 x \cdot - 5 = 0$

Step 1.3.1.8

Multiply $- 5$ by $- 22$ .

$- 12 x^{3} + 30 x^{2} - 44 x^{2} + 110 x = 0$

Step 1.3.2

Subtract $44 x^{2}$ from $30 x^{2}$ .

$- 12 x^{3} - 14 x^{2} + 110 x = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - \frac{11}{3}, 0, \frac{5}{2}$

Step 3