Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $6$ from both sides of the equation.

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

Step 1.2

Subtract $6$ from $- 18$ .

$6 n^{2} - 18 n - 24 = 0$

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 $a = 6$ , $b = - 18$ , and $c = - 24$ into the quadratic formula and solve for $n$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (6 \cdot - 24)}}{2 \cdot 6}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 18$ to the power of $2$ .

$n = \frac{18 \pm \sqrt{324 - 4 \cdot 6 \cdot - 24}}{2 \cdot 6}$

Step 4.1.2

Multiply $- 4 \cdot 6 \cdot - 24$ .

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Step 4.1.2.1

Multiply $- 4$ by $6$ .

$n = \frac{18 \pm \sqrt{324 - 24 \cdot - 24}}{2 \cdot 6}$

Step 4.1.2.2

Multiply $- 24$ by $- 24$ .

$n = \frac{18 \pm \sqrt{324 + 576}}{2 \cdot 6}$

Step 4.1.3

Add $324$ and $576$ .

$n = \frac{18 \pm \sqrt{900}}{2 \cdot 6}$

Step 4.1.4

Rewrite $900$ as $30^{2}$ .

$n = \frac{18 \pm \sqrt{30^{2}}}{2 \cdot 6}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$n = \frac{18 \pm 30}{2 \cdot 6}$

Step 4.2

Multiply $2$ by $6$ .

$n = \frac{18 \pm 30}{12}$

Step 4.3

Simplify $\frac{18 \pm 30}{12}$ .

$n = \frac{3 \pm 5}{2}$

Step 5

The final answer is the combination of both solutions.

$n = 4, - 1$