Solve for d d(4d+5)-8=0

$d (4 d + 5) - 8 = 0$

Step 1

Simplify each term.

Apply the distributive property.

$d (4 d) + d \cdot 5 - 8 = 0$

Rewrite using the commutative property of multiplication.

$4 d \cdot d + d \cdot 5 - 8 = 0$

Move $5$ to the left of $d$ .

$4 d \cdot d + 5 \cdot d - 8 = 0$

Multiply $d$ by $d$ by adding the exponents.

Move $d$ .

$4 (d \cdot d) + 5 \cdot d - 8 = 0$

Multiply $d$ by $d$ .

$4 d^{2} + 5 \cdot d - 8 = 0$

$4 d^{2} + 5 d - 8 = 0$

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 $a = 4$ , $b = 5$ , and $c = - 8$ into the quadratic formula and solve for $d$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot - 8)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

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

$d = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 8}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$d = \frac{- 5 \pm \sqrt{25 - 16 \cdot - 8}}{2 \cdot 4}$

Multiply $- 16$ by $- 8$ .

$d = \frac{- 5 \pm \sqrt{25 + 128}}{2 \cdot 4}$

Add $25$ and $128$ .

$d = \frac{- 5 \pm \sqrt{153}}{2 \cdot 4}$

Rewrite $153$ as $3^{2} \cdot 17$ .

Factor $9$ out of $153$ .

$d = \frac{- 5 \pm \sqrt{9 (17)}}{2 \cdot 4}$

Rewrite $9$ as $3^{2}$ .

$d = \frac{- 5 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 4}$

Pull terms out from under the radical.

$d = \frac{- 5 \pm 3 \sqrt{17}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$d = \frac{- 5 \pm 3 \sqrt{17}}{8}$

Step 5

The final answer is the combination of both solutions.

$d = - \frac{5 - 3 \sqrt{17}}{8}, - \frac{5 + 3 \sqrt{17}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{5 - 3 \sqrt{17}}{8}, - \frac{5 + 3 \sqrt{17}}{8}$

Decimal Form:

$d = 0.92116460 \dots, - 2.17116460 \dots$

Solve for d d(4d+5)-8=0

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