Algebra Examples

$N = \frac{t^{2} - t}{2}$

Step 1

Rewrite the equation as $\frac{t^{2} - t}{2} = N$ .

$\frac{t^{2} - t}{2} = N$

Step 2

Multiply both sides of the equation by $2$ .

$2 \frac{t^{2} - t}{2} = 2 N$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Cancel the common factor.

$2 \frac{t^{2} - t}{2} = 2 N$

Step 3.1.2

Rewrite the expression.

$t^{2} - t = 2 N$

Step 4

Subtract $2 N$ from both sides of the equation.

$t^{2} - t - 2 N = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1$ to the power of $2$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- 2 N)}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot (- 2 N)}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 2$ .

$t = \frac{1 \pm \sqrt{1 + 8 N}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$t = \frac{1 \pm \sqrt{1 + 8 N}}{2}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{1 + \sqrt{1 + 8 N}}{2}$

$t = \frac{1 - \sqrt{1 + 8 N}}{2}$