Pre-Algebra Examples

$6 n^{2} - 5 n - 12 = 18$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $18$ from both sides of the equation.

$6 n^{2} - 5 n - 12 - 18 = 0$

Step 1.2

Subtract $18$ from $- 12$ .

$6 n^{2} - 5 n - 30 = 0$

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 $a = 6$ , $b = - 5$ , and $c = - 30$ into the quadratic formula and solve for $n$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (6 \cdot - 30)}}{2 \cdot 6}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 5$ to the power of $2$ .

$n = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot - 30}}{2 \cdot 6}$

Step 4.1.2

Multiply $- 4 \cdot 6 \cdot - 30$ .

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Step 4.1.2.1

Multiply $- 4$ by $6$ .

$n = \frac{5 \pm \sqrt{25 - 24 \cdot - 30}}{2 \cdot 6}$

Step 4.1.2.2

Multiply $- 24$ by $- 30$ .

$n = \frac{5 \pm \sqrt{25 + 720}}{2 \cdot 6}$

Step 4.1.3

Add $25$ and $720$ .

$n = \frac{5 \pm \sqrt{745}}{2 \cdot 6}$

Step 4.2

Multiply $2$ by $6$ .

$n = \frac{5 \pm \sqrt{745}}{12}$

Step 5

The final answer is the combination of both solutions.

$n = \frac{5 + \sqrt{745}}{12}, \frac{5 - \sqrt{745}}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = \frac{5 + \sqrt{745}}{12}, \frac{5 - \sqrt{745}}{12}$

Decimal Form:

$n = 2.69122401 \dots, - 1.85789067 \dots$