Solve Using the Quadratic Formula 2d^2=8+10d

$2 d^{2} = 8 + 10 d$

Step 1

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

Subtract $8$ from both sides of the equation.

$2 d^{2} - 8 = 10 d$

Subtract $10 d$ from both sides of the equation.

$2 d^{2} - 8 - 10 d = 0$

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 $a = 2$ , $b = - 10$ , and $c = - 8$ into the quadratic formula and solve for $d$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot - 8)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

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

$d = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 8}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 8$ .

Multiply $- 4$ by $2$ .

$d = \frac{10 \pm \sqrt{100 - 8 \cdot - 8}}{2 \cdot 2}$

Multiply $- 8$ by $- 8$ .

$d = \frac{10 \pm \sqrt{100 + 64}}{2 \cdot 2}$

Add $100$ and $64$ .

$d = \frac{10 \pm \sqrt{164}}{2 \cdot 2}$

Rewrite $164$ as $2^{2} \cdot 41$ .

Factor $4$ out of $164$ .

$d = \frac{10 \pm \sqrt{4 (41)}}{2 \cdot 2}$

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

$d = \frac{10 \pm \sqrt{2^{2} \cdot 41}}{2 \cdot 2}$

Pull terms out from under the radical.

$d = \frac{10 \pm 2 \sqrt{41}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$d = \frac{10 \pm 2 \sqrt{41}}{4}$

Simplify $\frac{10 \pm 2 \sqrt{41}}{4}$ .

$d = \frac{5 \pm \sqrt{41}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$d = \frac{5 \pm \sqrt{41}}{2}$

Decimal Form:

$d = 5.70156211 \dots, - 0.70156211 \dots$