Algebra Examples

$5 t^{3} + 60 t^{2} = - 180 t$

Step 1

Add $180 t$ to both sides of the equation.

$5 t^{3} + 60 t^{2} + 180 t = 0$

Step 2

Factor the left side of the equation.

Tap for more steps...

Step 2.1

Factor $5 t$ out of $5 t^{3} + 60 t^{2} + 180 t$ .

Tap for more steps...

Step 2.1.1

Factor $5 t$ out of $5 t^{3}$ .

$5 t (t^{2}) + 60 t^{2} + 180 t = 0$

Step 2.1.2

Factor $5 t$ out of $60 t^{2}$ .

$5 t (t^{2}) + 5 t (12 t) + 180 t = 0$

Step 2.1.3

Factor $5 t$ out of $180 t$ .

$5 t (t^{2}) + 5 t (12 t) + 5 t (36) = 0$

Step 2.1.4

Factor $5 t$ out of $5 t (t^{2}) + 5 t (12 t)$ .

$5 t (t^{2} + 12 t) + 5 t (36) = 0$

Step 2.1.5

Factor $5 t$ out of $5 t (t^{2} + 12 t) + 5 t (36)$ .

$5 t (t^{2} + 12 t + 36) = 0$

Step 2.2

Factor using the perfect square rule.

Tap for more steps...

Step 2.2.1

Rewrite $36$ as $6^{2}$ .

$5 t (t^{2} + 12 t + 6^{2}) = 0$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 t = 2 \cdot t \cdot 6$

Step 2.2.3

Rewrite the polynomial.

$5 t (t^{2} + 2 \cdot t \cdot 6 + 6^{2}) = 0$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = t$ and $b = 6$ .

$5 t {(t + 6)}^{2} = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t = 0$

${(t + 6)}^{2} = 0$

Step 4

Set $t$ equal to $0$ .

$t = 0$

Step 5

Set ${(t + 6)}^{2}$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 5.1

Set ${(t + 6)}^{2}$ equal to $0$ .

${(t + 6)}^{2} = 0$

Step 5.2

Solve ${(t + 6)}^{2} = 0$ for $t$ .

Tap for more steps...

Step 5.2.1

Set the $t + 6$ equal to $0$ .

$t + 6 = 0$

Step 5.2.2

Subtract $6$ from both sides of the equation.

$t = - 6$

Step 6

The final solution is all the values that make $5 t {(t + 6)}^{2} = 0$ true.

$t = 0, - 6$