Solve for n 2n(4n+5)+6=0

$2 n (4 n + 5) + 6 = 0$

Step 1

Simplify each term.

Apply the distributive property.

$2 n (4 n) + 2 n \cdot 5 + 6 = 0$

Rewrite using the commutative property of multiplication.

$2 \cdot 4 n \cdot n + 2 n \cdot 5 + 6 = 0$

Multiply $5$ by $2$ .

$2 \cdot 4 n \cdot n + 10 n + 6 = 0$

Simplify each term.

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$2 \cdot 4 (n \cdot n) + 10 n + 6 = 0$

Multiply $n$ by $n$ .

$2 \cdot 4 n^{2} + 10 n + 6 = 0$

Multiply $2$ by $4$ .

$8 n^{2} + 10 n + 6 = 0$

Step 2

Factor $2$ out of $8 n^{2} + 10 n + 6$ .

Factor $2$ out of $8 n^{2}$ .

$2 (4 n^{2}) + 10 n + 6 = 0$

Factor $2$ out of $10 n$ .

$2 (4 n^{2}) + 2 (5 n) + 6 = 0$

Factor $2$ out of $6$ .

$2 (4 n^{2}) + 2 (5 n) + 2 (3) = 0$

Factor $2$ out of $2 (4 n^{2}) + 2 (5 n)$ .

$2 (4 n^{2} + 5 n) + 2 (3) = 0$

Factor $2$ out of $2 (4 n^{2} + 5 n) + 2 (3)$ .

$2 (4 n^{2} + 5 n + 3) = 0$

Step 3

Divide each term in $2 (4 n^{2} + 5 n + 3) = 0$ by $2$ and simplify.

Divide each term in $2 (4 n^{2} + 5 n + 3) = 0$ by $2$ .

$\frac{2 (4 n^{2} + 5 n + 3)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (4 n^{2} + 5 n + 3)}{2} = \frac{0}{2}$

Divide $4 n^{2} + 5 n + 3$ by $1$ .

$4 n^{2} + 5 n + 3 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$4 n^{2} + 5 n + 3 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = 5$ , and $c = 3$ into the quadratic formula and solve for $n$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot 3)}}{2 \cdot 4}$

Step 6

Simplify.

Simplify the numerator.

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

$n = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot 3}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 3$ .

Multiply $- 4$ by $4$ .

$n = \frac{- 5 \pm \sqrt{25 - 16 \cdot 3}}{2 \cdot 4}$

Multiply $- 16$ by $3$ .

$n = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 4}$

Subtract $48$ from $25$ .

$n = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 4}$

Rewrite $- 23$ as $- 1 (23)$ .

$n = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$n = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$n = \frac{- 5 \pm i \sqrt{23}}{8}$

Step 7

The final answer is the combination of both solutions.

$n = - \frac{5 - i \sqrt{23}}{8}, - \frac{5 + i \sqrt{23}}{8}$

Solve for n 2n(4n+5)+6=0

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