Question Determine if $t=-5$ is a solution to the equation $-2t^3 - 4t^2 + 4t - 3 = 127$ .

Studdy Solution

STEP 1

Assumptions
1. The given equation is $-2t^{3} - 4t^{2} + 4t - 3 = 127$ .
2. We need to determine if $t = -5$ is a solution to the equation.

STEP 2

To decide whether $t = -5$ is a solution, we need to substitute $t$ with $-5$ in the equation and check if the equality holds.

STEP 3

Substitute $t = -5$ into the equation.
$-2(-5)^{3} - 4(-5)^{2} + 4(-5) - 3$

STEP 4

First, calculate the cube of $-5$ .
$(-5)^{3} = -125$

STEP 5

Now, calculate the square of $-5$ .
$(-5)^{2} = 25$

STEP 6

Substitute these values into the equation.
$-2(-125) - 4(25) + 4(-5) - 3$

STEP 7

Multiply $-2$ by $-125$ .
$-2 \times -125 = 250$

STEP 8

Multiply $-4$ by $25$ .
$-4 \times 25 = -100$

STEP 9

Multiply $4$ by $-5$ .
$4 \times -5 = -20$

STEP 10

Now, add all the terms together.
$250 - 100 - 20 - 3$

STEP 11

First, subtract $100$ from $250$ .
$250 - 100 = 150$

STEP 12

Now, subtract $20$ from $150$ .
$150 - 20 = 130$

STEP 13

Finally, subtract $3$ from $130$ .
$130 - 3 = 127$

STEP 14

The result of the left-hand side of the equation after substituting $t = -5$ is $127$ , which is equal to the right-hand side of the original equation.

STEP 15

Since both sides of the equation are equal after the substitution, we can conclude that $t = -5$ is indeed a solution to the equation.
$-2 t^{3} - 4 t^{2} + 4 t - 3 = 127$

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