Expand the Trigonometric Expression (3-t)(2+3t)-(3t-1)^2

$(3 - t) (2 + 3 t) - {(3 t - 1)}^{2}$

Step 1

Simplify each term.

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

Apply the distributive property.

$3 (2 + 3 t) - t (2 + 3 t) - {(3 t - 1)}^{2}$

Apply the distributive property.

$3 \cdot 2 + 3 (3 t) - t (2 + 3 t) - {(3 t - 1)}^{2}$

Apply the distributive property.

$3 \cdot 2 + 3 (3 t) - t \cdot 2 - t (3 t) - {(3 t - 1)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $2$ .

$6 + 3 (3 t) - t \cdot 2 - t (3 t) - {(3 t - 1)}^{2}$

Multiply $3$ by $3$ .

$6 + 9 t - t \cdot 2 - t (3 t) - {(3 t - 1)}^{2}$

Multiply $2$ by $- 1$ .

$6 + 9 t - 2 t - t (3 t) - {(3 t - 1)}^{2}$

Rewrite using the commutative property of multiplication.

$6 + 9 t - 2 t - 1 \cdot 3 t \cdot t - {(3 t - 1)}^{2}$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$6 + 9 t - 2 t - 1 \cdot 3 (t \cdot t) - {(3 t - 1)}^{2}$

Multiply $t$ by $t$ .

$6 + 9 t - 2 t - 1 \cdot 3 t^{2} - {(3 t - 1)}^{2}$

Multiply $- 1$ by $3$ .

$6 + 9 t - 2 t - 3 t^{2} - {(3 t - 1)}^{2}$

Subtract $2 t$ from $9 t$ .

$6 + 7 t - 3 t^{2} - {(3 t - 1)}^{2}$

Rewrite ${(3 t - 1)}^{2}$ as $(3 t - 1) (3 t - 1)$ .

$6 + 7 t - 3 t^{2} - ((3 t - 1) (3 t - 1))$

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

Apply the distributive property.

$6 + 7 t - 3 t^{2} - (3 t (3 t - 1) - 1 (3 t - 1))$

Apply the distributive property.

$6 + 7 t - 3 t^{2} - (3 t (3 t) + 3 t \cdot - 1 - 1 (3 t - 1))$

Apply the distributive property.

$6 + 7 t - 3 t^{2} - (3 t (3 t) + 3 t \cdot - 1 - 1 (3 t) - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$6 + 7 t - 3 t^{2} - (3 \cdot 3 t \cdot t + 3 t \cdot - 1 - 1 (3 t) - 1 \cdot - 1)$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$6 + 7 t - 3 t^{2} - (3 \cdot 3 (t \cdot t) + 3 t \cdot - 1 - 1 (3 t) - 1 \cdot - 1)$

Multiply $t$ by $t$ .

$6 + 7 t - 3 t^{2} - (3 \cdot 3 t^{2} + 3 t \cdot - 1 - 1 (3 t) - 1 \cdot - 1)$

Multiply $3$ by $3$ .

$6 + 7 t - 3 t^{2} - (9 t^{2} + 3 t \cdot - 1 - 1 (3 t) - 1 \cdot - 1)$

Multiply $- 1$ by $3$ .

$6 + 7 t - 3 t^{2} - (9 t^{2} - 3 t - 1 (3 t) - 1 \cdot - 1)$

Multiply $3$ by $- 1$ .

$6 + 7 t - 3 t^{2} - (9 t^{2} - 3 t - 3 t - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$6 + 7 t - 3 t^{2} - (9 t^{2} - 3 t - 3 t + 1)$

Subtract $3 t$ from $- 3 t$ .

$6 + 7 t - 3 t^{2} - (9 t^{2} - 6 t + 1)$

Apply the distributive property.

$6 + 7 t - 3 t^{2} - (9 t^{2}) - (- 6 t) - 1 \cdot 1$

Simplify.

Multiply $9$ by $- 1$ .

$6 + 7 t - 3 t^{2} - 9 t^{2} - (- 6 t) - 1 \cdot 1$

Multiply $- 6$ by $- 1$ .

$6 + 7 t - 3 t^{2} - 9 t^{2} + 6 t - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$6 + 7 t - 3 t^{2} - 9 t^{2} + 6 t - 1$

Step 2

Simplify by adding terms.

Subtract $1$ from $6$ .

$7 t - 3 t^{2} - 9 t^{2} + 6 t + 5$

Add $7 t$ and $6 t$ .

$- 3 t^{2} - 9 t^{2} + 13 t + 5$

Subtract $9 t^{2}$ from $- 3 t^{2}$ .

$- 12 t^{2} + 13 t + 5$

Expand the Trigonometric Expression (3-t)(2+3t)-(3t-1)^2

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