Solve for d 5d^2+15d+5=-6

$5 d^{2} + 15 d + 5 = - 6$

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

Add $6$ to both sides of the equation.

$5 d^{2} + 15 d + 5 + 6 = 0$

Add $5$ and $6$ .

$5 d^{2} + 15 d + 11 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = 15$ , and $c = 11$ into the quadratic formula and solve for $d$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (5 \cdot 11)}}{2 \cdot 5}$

Simplify.

Simplify the numerator.

Raise $15$ to the power of $2$ .

$d = \frac{- 15 \pm \sqrt{225 - 4 \cdot 5 \cdot 11}}{2 \cdot 5}$

Multiply $- 4$ by $5$ .

$d = \frac{- 15 \pm \sqrt{225 - 20 \cdot 11}}{2 \cdot 5}$

Multiply $- 20$ by $11$ .

$d = \frac{- 15 \pm \sqrt{225 - 220}}{2 \cdot 5}$

Subtract $220$ from $225$ .

$d = \frac{- 15 \pm \sqrt{5}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$d = \frac{- 15 \pm \sqrt{5}}{10}$

The final answer is the combination of both solutions.

$d = - \frac{15 - \sqrt{5}}{10}, - \frac{15 + \sqrt{5}}{10}$

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{15 - \sqrt{5}}{10}, - \frac{15 + \sqrt{5}}{10}$

Decimal Form:

$d = - 1.27639320 \dots, - 1.72360679 \dots$