Trigonometry Examples

${(2 x + 3)}^{2} - {(2 x - 3)}^{2} = 24 x$

Step 1

Subtract $24 x$ from both sides of the equation.

${(2 x + 3)}^{2} - {(2 x - 3)}^{2} - 24 x = 0$

Step 2

Simplify ${(2 x + 3)}^{2} - {(2 x - 3)}^{2} - 24 x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(2 x + 3)}^{2}$ as $(2 x + 3) (2 x + 3)$ .

$(2 x + 3) (2 x + 3) - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.2

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$2 x (2 x + 3) + 3 (2 x + 3) - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3) - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.4

Multiply $3$ by $2$ .

$4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.5

Multiply $2$ by $3$ .

$4 x^{2} + 6 x + 6 x + 3 \cdot 3 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.1.6

Multiply $3$ by $3$ .

$4 x^{2} + 6 x + 6 x + 9 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.3.2

Add $6 x$ and $6 x$ .

$4 x^{2} + 12 x + 9 - {(2 x - 3)}^{2} - 24 x = 0$

Step 2.1.4

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

$4 x^{2} + 12 x + 9 - ((2 x - 3) (2 x - 3)) - 24 x = 0$

Step 2.1.5

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$4 x^{2} + 12 x + 9 - (2 x (2 x - 3) - 3 (2 x - 3)) - 24 x = 0$

Step 2.1.5.2

Apply the distributive property.

$4 x^{2} + 12 x + 9 - (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3)) - 24 x = 0$

Step 2.1.5.3

Apply the distributive property.

$4 x^{2} + 12 x + 9 - (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Rewrite using the commutative property of multiplication.

$4 x^{2} + 12 x + 9 - (2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.2

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

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Step 2.1.6.1.2.1

Move $x$ .

$4 x^{2} + 12 x + 9 - (2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.2.2

Multiply $x$ by $x$ .

$4 x^{2} + 12 x + 9 - (2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 12 x + 9 - (4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.4

Multiply $- 3$ by $2$ .

$4 x^{2} + 12 x + 9 - (4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.5

Multiply $2$ by $- 3$ .

$4 x^{2} + 12 x + 9 - (4 x^{2} - 6 x - 6 x - 3 \cdot - 3) - 24 x = 0$

Step 2.1.6.1.6

Multiply $- 3$ by $- 3$ .

$4 x^{2} + 12 x + 9 - (4 x^{2} - 6 x - 6 x + 9) - 24 x = 0$

Step 2.1.6.2

Subtract $6 x$ from $- 6 x$ .

$4 x^{2} + 12 x + 9 - (4 x^{2} - 12 x + 9) - 24 x = 0$

Step 2.1.7

Apply the distributive property.

$4 x^{2} + 12 x + 9 - (4 x^{2}) - (- 12 x) - 1 \cdot 9 - 24 x = 0$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $4$ by $- 1$ .

$4 x^{2} + 12 x + 9 - 4 x^{2} - (- 12 x) - 1 \cdot 9 - 24 x = 0$

Step 2.1.8.2

Multiply $- 12$ by $- 1$ .

$4 x^{2} + 12 x + 9 - 4 x^{2} + 12 x - 1 \cdot 9 - 24 x = 0$

Step 2.1.8.3

Multiply $- 1$ by $9$ .

$4 x^{2} + 12 x + 9 - 4 x^{2} + 12 x - 9 - 24 x = 0$

Step 2.2

Combine the opposite terms in $4 x^{2} + 12 x + 9 - 4 x^{2} + 12 x - 9 - 24 x$ .

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Step 2.2.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$12 x + 9 + 0 + 12 x - 9 - 24 x = 0$

Step 2.2.2

Add $12 x + 9$ and $0$ .

$12 x + 9 + 12 x - 9 - 24 x = 0$

Step 2.2.3

Subtract $9$ from $9$ .

$12 x + 12 x + 0 - 24 x = 0$

Step 2.2.4

Add $12 x + 12 x$ and $0$ .

$12 x + 12 x - 24 x = 0$

Step 2.3

Add $12 x$ and $12 x$ .

$24 x - 24 x = 0$

Step 2.4

Subtract $24 x$ from $24 x$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true.

Always true