Solve for t 4t(t-5)-2t(3t+2)+108=3t(2t-5)-t(8t-3)

$4 t (t - 5) - 2 t (3 t + 2) + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Step 1

Simplify $4 t (t - 5) - 2 t (3 t + 2) + 108$ .

Simplify each term.

Apply the distributive property.

$4 t \cdot t + 4 t \cdot - 5 - 2 t (3 t + 2) + 108 = 3 t (2 t - 5) - t (8 t - 3)$

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

Move $t$ .

$4 (t \cdot t) + 4 t \cdot - 5 - 2 t (3 t + 2) + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Multiply $t$ by $t$ .

$4 t^{2} + 4 t \cdot - 5 - 2 t (3 t + 2) + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Multiply $- 5$ by $4$ .

$4 t^{2} - 20 t - 2 t (3 t + 2) + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Apply the distributive property.

$4 t^{2} - 20 t - 2 t (3 t) - 2 t \cdot 2 + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Rewrite using the commutative property of multiplication.

$4 t^{2} - 20 t - 2 \cdot 3 t \cdot t - 2 t \cdot 2 + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Multiply $2$ by $- 2$ .

$4 t^{2} - 20 t - 2 \cdot 3 t \cdot t - 4 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Simplify each term.

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

Move $t$ .

$4 t^{2} - 20 t - 2 \cdot 3 (t \cdot t) - 4 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Multiply $t$ by $t$ .

$4 t^{2} - 20 t - 2 \cdot 3 t^{2} - 4 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Multiply $- 2$ by $3$ .

$4 t^{2} - 20 t - 6 t^{2} - 4 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Simplify by adding terms.

Subtract $6 t^{2}$ from $4 t^{2}$ .

$- 2 t^{2} - 20 t - 4 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Subtract $4 t$ from $- 20 t$ .

$- 2 t^{2} - 24 t + 108 = 3 t (2 t - 5) - t (8 t - 3)$

Step 2

Simplify $3 t (2 t - 5) - t (8 t - 3)$ .

Simplify each term.

Apply the distributive property.

$- 2 t^{2} - 24 t + 108 = 3 t (2 t) + 3 t \cdot - 5 - t (8 t - 3)$

Rewrite using the commutative property of multiplication.

$- 2 t^{2} - 24 t + 108 = 3 \cdot 2 t \cdot t + 3 t \cdot - 5 - t (8 t - 3)$

Multiply $- 5$ by $3$ .

$- 2 t^{2} - 24 t + 108 = 3 \cdot 2 t \cdot t - 15 t - t (8 t - 3)$

Simplify each term.

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

Move $t$ .

$- 2 t^{2} - 24 t + 108 = 3 \cdot 2 (t \cdot t) - 15 t - t (8 t - 3)$

Multiply $t$ by $t$ .

$- 2 t^{2} - 24 t + 108 = 3 \cdot 2 t^{2} - 15 t - t (8 t - 3)$

Multiply $3$ by $2$ .

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - t (8 t - 3)$

Apply the distributive property.

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - t (8 t) - t \cdot - 3$

Rewrite using the commutative property of multiplication.

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - 1 \cdot 8 t \cdot t - t \cdot - 3$

Multiply $- 3$ by $- 1$ .

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - 1 \cdot 8 t \cdot t + 3 t$

Simplify each term.

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

Move $t$ .

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - 1 \cdot 8 (t \cdot t) + 3 t$

Multiply $t$ by $t$ .

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - 1 \cdot 8 t^{2} + 3 t$

Multiply $- 1$ by $8$ .

$- 2 t^{2} - 24 t + 108 = 6 t^{2} - 15 t - 8 t^{2} + 3 t$

Simplify by adding terms.

Subtract $8 t^{2}$ from $6 t^{2}$ .

$- 2 t^{2} - 24 t + 108 = - 2 t^{2} - 15 t + 3 t$

Add $- 15 t$ and $3 t$ .

$- 2 t^{2} - 24 t + 108 = - 2 t^{2} - 12 t$

Step 3

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

Add $2 t^{2}$ to both sides of the equation.

$- 2 t^{2} - 24 t + 108 + 2 t^{2} = - 12 t$

Add $12 t$ to both sides of the equation.

$- 2 t^{2} - 24 t + 108 + 2 t^{2} + 12 t = 0$

Combine the opposite terms in $- 2 t^{2} - 24 t + 108 + 2 t^{2} + 12 t$ .

Add $- 2 t^{2}$ and $2 t^{2}$ .

$- 24 t + 108 + 0 + 12 t = 0$

Add $- 24 t + 108$ and $0$ .

$- 24 t + 108 + 12 t = 0$

Add $- 24 t$ and $12 t$ .

$- 12 t + 108 = 0$

Step 4

Subtract $108$ from both sides of the equation.

$- 12 t = - 108$

Step 5

Divide each term in $- 12 t = - 108$ by $- 12$ and simplify.

Divide each term in $- 12 t = - 108$ by $- 12$ .

$\frac{- 12 t}{- 12} = \frac{- 108}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 t}{- 12} = \frac{- 108}{- 12}$

Divide $t$ by $1$ .

$t = \frac{- 108}{- 12}$

Simplify the right side.

Divide $- 108$ by $- 12$ .

$t = 9$

Solve for t 4t(t-5)-2t(3t+2)+108=3t(2t-5)-t(8t-3)

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