Solve for w A=-4w^2+64w

$A = - 4 w^{2} + 64 w$

Step 1

Rewrite the equation as $- 4 w^{2} + 64 w = A$ .

$- 4 w^{2} + 64 w = A$

Step 2

Subtract $A$ from both sides of the equation.

$- 4 w^{2} + 64 w - A = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 4$ , $b = 64$ , and $c = - A$ into the quadratic formula and solve for $w$ .

$\frac{- 64 \pm \sqrt{64^{2} - 4 \cdot (- 4 \cdot (- A))}}{2 \cdot - 4}$

Step 5

Simplify.

Simplify the numerator.

Raise $64$ to the power of $2$ .

$w = \frac{- 64 \pm \sqrt{4096 - 4 \cdot - 4 \cdot (- 1 A)}}{2 \cdot - 4}$

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

Multiply $- 4$ by $- 4$ .

$w = \frac{- 64 \pm \sqrt{4096 + 16 \cdot (- 1 A)}}{2 \cdot - 4}$

Multiply $16$ by $- 1$ .

$w = \frac{- 64 \pm \sqrt{4096 - 16 A}}{2 \cdot - 4}$

Factor $16$ out of $4096 - 16 A$ .

Factor $16$ out of $4096$ .

$w = \frac{- 64 \pm \sqrt{16 (256) - 16 A}}{2 \cdot - 4}$

Factor $16$ out of $- 16 A$ .

$w = \frac{- 64 \pm \sqrt{16 (256) + 16 (- A)}}{2 \cdot - 4}$

Factor $16$ out of $16 (256) + 16 (- A)$ .

$w = \frac{- 64 \pm \sqrt{16 (256 - A)}}{2 \cdot - 4}$

Rewrite $16 (256 - A)$ as ${(2^{2})}^{2} (256 - A)$ .

Rewrite $16$ as $4^{2}$ .

$w = \frac{- 64 \pm \sqrt{4^{2} (256 - A)}}{2 \cdot - 4}$

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

$w = \frac{- 64 \pm \sqrt{{(2^{2})}^{2} (256 - A)}}{2 \cdot - 4}$

Pull terms out from under the radical.

$w = \frac{- 64 \pm 2^{2} \sqrt{256 - A}}{2 \cdot - 4}$

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

$w = \frac{- 64 \pm 4 \sqrt{256 - A}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$w = \frac{- 64 \pm 4 \sqrt{256 - A}}{- 8}$

Simplify $\frac{- 64 \pm 4 \sqrt{256 - A}}{- 8}$ .

$w = \frac{16 \pm \sqrt{256 - A}}{2}$

Step 6

The final answer is the combination of both solutions.

$w = \frac{16 + \sqrt{256 - A}}{2}$

$w = \frac{16 - \sqrt{256 - A}}{2}$

Solve for w A=-4w^2+64w

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