Precalculus Examples

$s = \frac{n (n + 1)}{7}$

Step 1

Rewrite the equation as $\frac{n (n + 1)}{7} = s$ .

$\frac{n (n + 1)}{7} = s$

Step 2

Multiply both sides of the equation by $7$ .

$7 \frac{n (n + 1)}{7} = 7 s$

Step 3

Simplify the left side.

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Step 3.1

Simplify $7 \frac{n (n + 1)}{7}$ .

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Step 3.1.1

Cancel the common factor of $7$ .

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Step 3.1.1.1

Cancel the common factor.

$7 \frac{n (n + 1)}{7} = 7 s$

Step 3.1.1.2

Rewrite the expression.

$n (n + 1) = 7 s$

Step 3.1.2

Apply the distributive property.

$n \cdot n + n \cdot 1 = 7 s$

Step 3.1.3

Simplify the expression.

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Step 3.1.3.1

Multiply $n$ by $n$ .

$n^{2} + n \cdot 1 = 7 s$

Step 3.1.3.2

Multiply $n$ by $1$ .

$n^{2} + n = 7 s$

Step 4

Subtract $7 s$ from both sides of the equation.

$n^{2} + n - 7 s = 0$

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 $c = - 7 s$ into the quadratic formula and solve for $n$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot (- 7 s))}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$n = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- 7 s)}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{- 1 \pm \sqrt{1 - 4 \cdot (- 7 s)}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 7$ .

$n = \frac{- 1 \pm \sqrt{1 + 28 s}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$n = \frac{- 1 \pm \sqrt{1 + 28 s}}{2}$

Step 8

The final answer is the combination of both solutions.

$n = - \frac{1 - \sqrt{1 + 28 s}}{2}$

$n = - \frac{1 + \sqrt{28 s + 1}}{2}$