Solve Using the Quadratic Formula 3n^2+10n=5

$3 n^{2} + 10 n = 5$

Step 1

Subtract $5$ from both sides of the equation.

$3 n^{2} + 10 n - 5 = 0$

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 $a = 3$ , $b = 10$ , and $c = - 5$ into the quadratic formula and solve for $n$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (3 \cdot - 5)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$n = \frac{- 10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$n = \frac{- 10 \pm \sqrt{100 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$n = \frac{- 10 \pm \sqrt{100 + 60}}{2 \cdot 3}$

Add $100$ and $60$ .

$n = \frac{- 10 \pm \sqrt{160}}{2 \cdot 3}$

Rewrite $160$ as $4^{2} \cdot 10$ .

Factor $16$ out of $160$ .

$n = \frac{- 10 \pm \sqrt{16 (10)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$n = \frac{- 10 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 3}$

Pull terms out from under the radical.

$n = \frac{- 10 \pm 4 \sqrt{10}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$n = \frac{- 10 \pm 4 \sqrt{10}}{6}$

Simplify $\frac{- 10 \pm 4 \sqrt{10}}{6}$ .

$n = \frac{- 5 \pm 2 \sqrt{10}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{- 5 \pm 2 \sqrt{10}}{3}$

Decimal Form:

$n = 0.44151844 \dots, - 3.77485177 \dots$