Algebra Examples

$2.5 n^{2} - 6.5 n - 33$

Step 1

Set $2.5 n^{2} - 6.5 n - 33$ equal to $0$ .

$2.5 n^{2} - 6.5 n - 33 = 0$

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 $a = 2.5$ , $b = - 6.5$ , and $c = - 33$ into the quadratic formula and solve for $n$ .

$\frac{6.5 \pm \sqrt{{(- 6.5)}^{2} - 4 \cdot (2.5 \cdot - 33)}}{2 \cdot 2.5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6.5$ to the power of $2$ .

$n = \frac{6.5 \pm \sqrt{42.25 - 4 \cdot 2.5 \cdot - 33}}{2 \cdot 2.5}$

Step 4.1.2

Multiply $- 4 \cdot 2.5 \cdot - 33$ .

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Step 4.1.2.1

Multiply $- 4$ by $2.5$ .

$n = \frac{6.5 \pm \sqrt{42.25 - 10 \cdot - 33}}{2 \cdot 2.5}$

Step 4.1.2.2

Multiply $- 10$ by $- 33$ .

$n = \frac{6.5 \pm \sqrt{42.25 + 330}}{2 \cdot 2.5}$

Step 4.1.3

Add $42.25$ and $330$ .

$n = \frac{6.5 \pm \sqrt{372.25}}{2 \cdot 2.5}$

Step 4.2

Multiply $2$ by $2.5$ .

$n = \frac{6.5 \pm \sqrt{372.25}}{5}$

Step 4.3

Multiply by $1$ .

$n = \frac{1 (6.5 \pm \sqrt{372.25})}{5}$

Step 4.4

Factor $5$ out of $5$ .

$n = \frac{1 (6.5 \pm \sqrt{372.25})}{5 (1)}$

Step 4.5

Separate fractions.

$n = \frac{1}{5} \cdot \frac{6.5 \pm \sqrt{372.25}}{1}$

Step 4.6

Divide $1$ by $5$ .

$n = 0.2 (\frac{6.5 \pm \sqrt{372.25}}{1})$

Step 4.7

Cancel the common factor of $0.2$ .

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Step 4.7.1

Factor $0.2$ out of $1$ .

$n = 0.2 (\frac{6.5 \pm \sqrt{372.25}}{0.2 \cdot 5})$

Step 4.7.2

Cancel the common factor.

$n = 0.2 (\frac{6.5 \pm \sqrt{372.25}}{0.2 \cdot 5})$

Step 4.7.3

Rewrite the expression.

$n = \frac{6.5 \pm \sqrt{372.25}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{6.5 \pm \sqrt{372.25}}{5}$

Decimal Form:

$n = 5.15875627 \dots, - 2.55875627 \dots$