Algebra Examples

$S (t) = \frac{1}{3} t^{3} - 3.5 t^{2} + 10 t$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{1}{3} t^{3} - 3.5 t^{2} + 10 t = S t$

Step 2

Combine $\frac{1}{3}$ and $t^{3}$ .

$\frac{t^{3}}{3} - 3.5 t^{2} + 10 t = S t$

Step 3

Subtract $S t$ from both sides of the equation.

$\frac{t^{3}}{3} - 3.5 t^{2} + 10 t - S t = 0$

Step 4

Factor $t$ out of $\frac{t^{3}}{3} - 3.5 t^{2} + 10 t - S t$ .

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Step 4.1

Factor $t$ out of $\frac{t^{3}}{3}$ .

$t (\frac{t^{2}}{3}) - 3.5 t^{2} + 10 t - S t = 0$

Step 4.2

Factor $t$ out of $- 3.5 t^{2}$ .

$t (\frac{t^{2}}{3}) + t (- 3.5 t) + 10 t - S t = 0$

Step 4.3

Factor $t$ out of $10 t$ .

$t (\frac{t^{2}}{3}) + t (- 3.5 t) + t \cdot 10 - S t = 0$

Step 4.4

Factor $t$ out of $- S t$ .

$t (\frac{t^{2}}{3}) + t (- 3.5 t) + t \cdot 10 + t (- S) = 0$

Step 4.5

Factor $t$ out of $t \frac{t^{2}}{3} + t (- 3.5 t)$ .

$t (\frac{t^{2}}{3} - 3.5 t) + t \cdot 10 + t (- S) = 0$

Step 4.6

Factor $t$ out of $t (\frac{t^{2}}{3} - 3.5 t) + t \cdot 10$ .

$t (\frac{t^{2}}{3} - 3.5 t + 10) + t (- S) = 0$

Step 4.7

Factor $t$ out of $t (\frac{t^{2}}{3} - 3.5 t + 10) + t (- S)$ .

$t (\frac{t^{2}}{3} - 3.5 t + 10 - S) = 0$

Step 5

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

$t = 0$

$\frac{t^{2}}{3} - 3.5 t + 10 - S = 0$

Step 6

Set $t$ equal to $0$ .

$t = 0$

Step 7

Set $\frac{t^{2}}{3} - 3.5 t + 10 - S$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $\frac{t^{2}}{3} - 3.5 t + 10 - S$ equal to $0$ .

$\frac{t^{2}}{3} - 3.5 t + 10 - S = 0$

Step 7.2

Solve $\frac{t^{2}}{3} - 3.5 t + 10 - S = 0$ for $t$ .

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Step 7.2.1

Multiply through by the least common denominator $3$ , then simplify.

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Step 7.2.1.1

Apply the distributive property.

$3 (\frac{t^{2}}{3}) + 3 (- 3.5 t) + 3 \cdot 10 + 3 (- S) = 0$

Step 7.2.1.2

Simplify.

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Step 7.2.1.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.2.1.1

Cancel the common factor.

$3 (\frac{t^{2}}{3}) + 3 (- 3.5 t) + 3 \cdot 10 + 3 (- S) = 0$

Step 7.2.1.2.1.2

Rewrite the expression.

$t^{2} + 3 (- 3.5 t) + 3 \cdot 10 + 3 (- S) = 0$

Step 7.2.1.2.2

Multiply $- 3.5$ by $3$ .

$t^{2} - 10.5 t + 3 \cdot 10 + 3 (- S) = 0$

Step 7.2.1.2.3

Multiply $3$ by $10$ .

$t^{2} - 10.5 t + 30 + 3 (- S) = 0$

Step 7.2.1.2.4

Multiply $- 1$ by $3$ .

$t^{2} - 10.5 t + 30 - 3 S = 0$

Step 7.2.1.3

Move $30$ .

$t^{2} - 10.5 t - 3 S + 30 = 0$

Step 7.2.1.4

Move $- 10.5 t$ .

$t^{2} - 3 S - 10.5 t + 30 = 0$

Step 7.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.3

Substitute the values $a = 1$ , $b = - 10.5$ , and $c = - 3 S + 30$ into the quadratic formula and solve for $t$ .

$\frac{10.5 \pm \sqrt{{(- 10.5)}^{2} - 4 \cdot (1 \cdot (- 3 S + 30))}}{2 \cdot 1}$

Step 7.2.4

Simplify.

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Step 7.2.4.1

Simplify the numerator.

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Step 7.2.4.1.1

Raise $- 10.5$ to the power of $2$ .

$t = \frac{10.5 \pm \sqrt{110.25 - 4 \cdot 1 \cdot (- 3 S + 30)}}{2 \cdot 1}$

Step 7.2.4.1.2

Multiply $- 4$ by $1$ .

$t = \frac{10.5 \pm \sqrt{110.25 - 4 \cdot (- 3 S + 30)}}{2 \cdot 1}$

Step 7.2.4.1.3

Apply the distributive property.

$t = \frac{10.5 \pm \sqrt{110.25 - 4 (- 3 S) - 4 \cdot 30}}{2 \cdot 1}$

Step 7.2.4.1.4

Multiply $- 3$ by $- 4$ .

$t = \frac{10.5 \pm \sqrt{110.25 + 12 S - 4 \cdot 30}}{2 \cdot 1}$

Step 7.2.4.1.5

Multiply $- 4$ by $30$ .

$t = \frac{10.5 \pm \sqrt{110.25 + 12 S - 120}}{2 \cdot 1}$

Step 7.2.4.1.6

Subtract $120$ from $110.25$ .

$t = \frac{10.5 \pm \sqrt{12 S - 9.75}}{2 \cdot 1}$

Step 7.2.4.1.7

Factor $0.75$ out of $12 S - 9.75$ .

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Step 7.2.4.1.7.1

Factor $0.75$ out of $12 S$ .

$t = \frac{10.5 \pm \sqrt{0.75 (16 S) - 9.75}}{2 \cdot 1}$

Step 7.2.4.1.7.2

Factor $0.75$ out of $- 9.75$ .

$t = \frac{10.5 \pm \sqrt{0.75 (16 S) + 0.75 (- 13)}}{2 \cdot 1}$

Step 7.2.4.1.7.3

Factor $0.75$ out of $0.75 (16 S) + 0.75 (- 13)$ .

$t = \frac{10.5 \pm \sqrt{0.75 (16 S - 13)}}{2 \cdot 1}$

Step 7.2.4.1.8

Rewrite $0.75 (16 S - 13)$ as ${({0.93060485}^{2})}^{2} (16 S - 13)$ .

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Step 7.2.4.1.8.1

Rewrite $0.75$ as ${0.8660254}^{2}$ .

$t = \frac{10.5 \pm \sqrt{{0.8660254}^{2} (16 S - 13)}}{2 \cdot 1}$

Step 7.2.4.1.8.2

Rewrite $0.8660254$ as ${0.93060485}^{2}$ .

$t = \frac{10.5 \pm \sqrt{{({0.93060485}^{2})}^{2} (16 S - 13)}}{2 \cdot 1}$

Step 7.2.4.1.9

Pull terms out from under the radical.

$t = \frac{10.5 \pm {0.93060485}^{2} \sqrt{16 S - 13}}{2 \cdot 1}$

Step 7.2.4.1.10

Raise $0.93060485$ to the power of $2$ .

$t = \frac{10.5 \pm 0.8660254 \sqrt{16 S - 13}}{2 \cdot 1}$

Step 7.2.4.2

Multiply $2$ by $1$ .

$t = \frac{10.5 \pm 0.8660254 \sqrt{16 S - 13}}{2}$

Step 7.2.5

The final answer is the combination of both solutions.

$t = \frac{10.5 + 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 - 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 + 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 - 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 + 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 - 0.8660254 \sqrt{16 S - 13}}{2}$

Step 8

The final solution is all the values that make $t (\frac{t^{2}}{3} - 3.5 t + 10 - S) = 0$ true.

$t = 0$

$t = \frac{10.5 + 0.8660254 \sqrt{16 S - 13}}{2}$

$t = \frac{10.5 - 0.8660254 \sqrt{16 S - 13}}{2}$