Calculus Examples

$w^{2} = 9 (w - 4)$

Step 1

Simplify $9 (w - 4)$ .

Tap for more steps...

Step 1.1

Apply the distributive property.

$w^{2} = 9 w + 9 \cdot - 4$

Step 1.2

Multiply $9$ by $- 4$ .

$w^{2} = 9 w - 36$

Step 2

Subtract $9 w$ from both sides of the equation.

$w^{2} - 9 w = - 36$

Step 3

Add $36$ to both sides of the equation.

$w^{2} - 9 w + 36 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 9$ , and $c = 36$ into the quadratic formula and solve for $w$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 36)}}{2 \cdot 1}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Raise $- 9$ to the power of $2$ .

$w = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $1$ .

$w = \frac{9 \pm \sqrt{81 - 4 \cdot 36}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $36$ .

$w = \frac{9 \pm \sqrt{81 - 144}}{2 \cdot 1}$

Step 6.1.3

Subtract $144$ from $81$ .

$w = \frac{9 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$w = \frac{9 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$w = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$w = \frac{9 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 6.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

Tap for more steps...

Step 6.1.7.1

Factor $9$ out of $63$ .

$w = \frac{9 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 6.1.7.2

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

$w = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$w = \frac{9 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 6.1.9

Move $3$ to the left of $i$ .

$w = \frac{9 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$w = \frac{9 \pm 3 i \sqrt{7}}{2}$

Step 7

The final answer is the combination of both solutions.

$w = \frac{9 + 3 i \sqrt{7}}{2}, \frac{9 - 3 i \sqrt{7}}{2}$