Algebra Examples

$t (2 t + 1) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $t (2 t + 1)$ .

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Step 1.1.1

Apply the distributive property.

$t (2 t) + t \cdot 1 = 0$

Step 1.1.2

Simplify the expression.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$2 t \cdot t + t \cdot 1 = 0$

Step 1.1.2.2

Multiply $t$ by $1$ .

$2 t \cdot t + t = 0$

Step 1.1.3

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

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Step 1.1.3.1

Move $t$ .

$2 (t \cdot t) + t = 0$

Step 1.1.3.2

Multiply $t$ by $t$ .

$2 t^{2} + t = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = 1$ , and $c = 0$ into the quadratic formula and solve for $t$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 0)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$t = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 0}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 0$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$t = \frac{- 1 \pm \sqrt{1 - 8 \cdot 0}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $0$ .

$t = \frac{- 1 \pm \sqrt{1 + 0}}{2 \cdot 2}$

Step 4.1.3

Add $1$ and $0$ .

$t = \frac{- 1 \pm \sqrt{1}}{2 \cdot 2}$

Step 4.1.4

Any root of $1$ is $1$ .

$t = \frac{- 1 \pm 1}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$t = \frac{- 1 \pm 1}{4}$

Step 5

The final answer is the combination of both solutions.

$t = 0, - \frac{1}{2}$