Solve for A P=0.006A^2-0.02A+120

$P = 0.006 A^{2} - 0.02 A + 120$

Step 1

Rewrite the equation as $0.006 A^{2} - 0.02 A + 120 = P$ .

$0.006 A^{2} - 0.02 A + 120 = P$

Step 2

Subtract $P$ from both sides of the equation.

$0.006 A^{2} - 0.02 A + 120 - P = 0$

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 $a = 0.006$ , $b = - 0.02$ , and $c = 120 - P$ into the quadratic formula and solve for $A$ .

$\frac{0.02 \pm \sqrt{{(- 0.02)}^{2} - 4 \cdot (0.006 \cdot (120 - P))}}{2 \cdot 0.006}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 0.02$ to the power of $2$ .

$A = \frac{0.02 \pm \sqrt{0.0004 - 4 \cdot 0.006 \cdot (120 - P)}}{2 \cdot 0.006}$

Multiply $- 4$ by $0.006$ .

$A = \frac{0.02 \pm \sqrt{0.0004 - 0.024 \cdot (120 - P)}}{2 \cdot 0.006}$

Apply the distributive property.

$A = \frac{0.02 \pm \sqrt{0.0004 - 0.024 \cdot 120 - 0.024 (- P)}}{2 \cdot 0.006}$

Multiply $- 0.024$ by $120$ .

$A = \frac{0.02 \pm \sqrt{0.0004 - 2.88 - 0.024 (- P)}}{2 \cdot 0.006}$

Multiply $- 1$ by $- 0.024$ .

$A = \frac{0.02 \pm \sqrt{0.0004 - 2.88 + 0.024 P}}{2 \cdot 0.006}$

Subtract $2.88$ from $0.0004$ .

$A = \frac{0.02 \pm \sqrt{- 2.8796 + 0.024 P}}{2 \cdot 0.006}$

Factor $0.0004$ out of $- 2.8796 + 0.024 P$ .

Factor $0.0004$ out of $- 2.8796$ .

$A = \frac{0.02 \pm \sqrt{0.0004 (- 7199) + 0.024 P}}{2 \cdot 0.006}$

Factor $0.0004$ out of $0.024 P$ .

$A = \frac{0.02 \pm \sqrt{0.0004 (- 7199) + 0.0004 (60 P)}}{2 \cdot 0.006}$

Factor $0.0004$ out of $0.0004 (- 7199) + 0.0004 (60 P)$ .

$A = \frac{0.02 \pm \sqrt{0.0004 (- 7199 + 60 P)}}{2 \cdot 0.006}$

Rewrite $0.0004 (- 7199 + 60 P)$ as ${({0.14142135}^{2})}^{2} (- 7199 + 60 P)$ .

Rewrite $0.0004$ as ${0.02}^{2}$ .

$A = \frac{0.02 \pm \sqrt{{0.02}^{2} (- 7199 + 60 P)}}{2 \cdot 0.006}$

Rewrite $0.02$ as ${0.14142135}^{2}$ .

$A = \frac{0.02 \pm \sqrt{{({0.14142135}^{2})}^{2} (- 7199 + 60 P)}}{2 \cdot 0.006}$

Pull terms out from under the radical.

$A = \frac{0.02 \pm {0.14142135}^{2} \sqrt{- 7199 + 60 P}}{2 \cdot 0.006}$

Raise $0.14142135$ to the power of $2$ .

$A = \frac{0.02 \pm 0.02 \sqrt{- 7199 + 60 P}}{2 \cdot 0.006}$

Multiply $2$ by $0.006$ .

$A = \frac{0.02 \pm 0.02 \sqrt{- 7199 + 60 P}}{0.012}$

Simplify $\frac{0.02 \pm 0.02 \sqrt{- 7199 + 60 P}}{0.012}$ .

$A = \frac{5 \pm 5 \sqrt{- 7199 + 60 P}}{3}$

Step 6

The final answer is the combination of both solutions.

$A = \frac{5 + 5 \sqrt{- 7199 + 60 P}}{3}$

$A = \frac{5 - 5 \sqrt{- 7199 + 60 P}}{3}$

Solve for A P=0.006A^2-0.02A+120

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