Solve for w w(2w-11)-8=0

$w (2 w - 11) - 8 = 0$

Step 1

Simplify each term.

Apply the distributive property.

$w (2 w) + w \cdot - 11 - 8 = 0$

Rewrite using the commutative property of multiplication.

$2 w \cdot w + w \cdot - 11 - 8 = 0$

Move $- 11$ to the left of $w$ .

$2 w \cdot w - 11 \cdot w - 8 = 0$

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$2 (w \cdot w) - 11 \cdot w - 8 = 0$

Multiply $w$ by $w$ .

$2 w^{2} - 11 \cdot w - 8 = 0$

$2 w^{2} - 11 w - 8 = 0$

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 $a = 2$ , $b = - 11$ , and $c = - 8$ into the quadratic formula and solve for $w$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (2 \cdot - 8)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$w = \frac{11 \pm \sqrt{121 - 4 \cdot 2 \cdot - 8}}{2 \cdot 2}$

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

Multiply $- 4$ by $2$ .

$w = \frac{11 \pm \sqrt{121 - 8 \cdot - 8}}{2 \cdot 2}$

Multiply $- 8$ by $- 8$ .

$w = \frac{11 \pm \sqrt{121 + 64}}{2 \cdot 2}$

Add $121$ and $64$ .

$w = \frac{11 \pm \sqrt{185}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$w = \frac{11 \pm \sqrt{185}}{4}$

Step 5

The final answer is the combination of both solutions.

$w = \frac{11 + \sqrt{185}}{4}, \frac{11 - \sqrt{185}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = \frac{11 + \sqrt{185}}{4}, \frac{11 - \sqrt{185}}{4}$

Decimal Form:

$w = 6.15036762 \dots, - 0.65036762 \dots$