Solve Using the Quadratic Formula 4n^2=-10n-3

$4 n^{2} = - 10 n - 3$

Step 1

Move all the expressions to the left side of the equation.

Add $10 n$ to both sides of the equation.

$4 n^{2} + 10 n = - 3$

Add $3$ to both sides of the equation.

$4 n^{2} + 10 n + 3 = 0$

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 $a = 4$ , $b = 10$ , and $c = 3$ into the quadratic formula and solve for $n$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $4$ .

$n = \frac{- 10 \pm \sqrt{100 - 16 \cdot 3}}{2 \cdot 4}$

Multiply $- 16$ by $3$ .

$n = \frac{- 10 \pm \sqrt{100 - 48}}{2 \cdot 4}$

Subtract $48$ from $100$ .

$n = \frac{- 10 \pm \sqrt{52}}{2 \cdot 4}$

Rewrite $52$ as $2^{2} \cdot 13$ .

Factor $4$ out of $52$ .

$n = \frac{- 10 \pm \sqrt{4 (13)}}{2 \cdot 4}$

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

$n = \frac{- 10 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 4}$

Pull terms out from under the radical.

$n = \frac{- 10 \pm 2 \sqrt{13}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$n = \frac{- 10 \pm 2 \sqrt{13}}{8}$

Simplify $\frac{- 10 \pm 2 \sqrt{13}}{8}$ .

$n = \frac{- 5 \pm \sqrt{13}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{- 5 \pm \sqrt{13}}{4}$

Decimal Form:

$n = - 0.34861218 \dots, - 2.15138781 \dots$