Solve 1300a=150000-9600a+288000/a

$$1300a = 150000- \frac{ 9600a+288000 }{ a }$$

Solve for a

$$a=\frac{6\sqrt{12649}}{13}+54\approx 105.908202998$$

$$a=-\frac{6\sqrt{12649}}{13}+54\approx 2.091797002$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$1300aa=a\times 150000-\left(9600a+288000\right)$$

Multiply a and a to get a^{2}.

$$1300a^{2}=a\times 150000-\left(9600a+288000\right)$$

To find the opposite of 9600a+288000, find the opposite of each term.

$$1300a^{2}=a\times 150000-9600a-288000$$

Combine a\times 150000 and -9600a to get 140400a.

$$1300a^{2}=140400a-288000$$

Subtract 140400a from both sides.

$$1300a^{2}-140400a=-288000$$

Add 288000 to both sides.

$$1300a^{2}-140400a+288000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1300 for a, -140400 for b, and 288000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-140400\right)±\sqrt{\left(-140400\right)^{2}-4\times 1300\times 288000}}{2\times 1300}$$

Square -140400.

$$a=\frac{-\left(-140400\right)±\sqrt{19712160000-4\times 1300\times 288000}}{2\times 1300}$$

Multiply -4 times 1300.

$$a=\frac{-\left(-140400\right)±\sqrt{19712160000-5200\times 288000}}{2\times 1300}$$

Multiply -5200 times 288000.

$$a=\frac{-\left(-140400\right)±\sqrt{19712160000-1497600000}}{2\times 1300}$$

Add 19712160000 to -1497600000.

$$a=\frac{-\left(-140400\right)±\sqrt{18214560000}}{2\times 1300}$$

Take the square root of 18214560000.

$$a=\frac{-\left(-140400\right)±1200\sqrt{12649}}{2\times 1300}$$

The opposite of -140400 is 140400.

$$a=\frac{140400±1200\sqrt{12649}}{2\times 1300}$$

Multiply 2 times 1300.

$$a=\frac{140400±1200\sqrt{12649}}{2600}$$

Now solve the equation a=\frac{140400±1200\sqrt{12649}}{2600} when ± is plus. Add 140400 to 1200\sqrt{12649}.

$$a=\frac{1200\sqrt{12649}+140400}{2600}$$

Divide 140400+1200\sqrt{12649} by 2600.

$$a=\frac{6\sqrt{12649}}{13}+54$$

Now solve the equation a=\frac{140400±1200\sqrt{12649}}{2600} when ± is minus. Subtract 1200\sqrt{12649} from 140400.

$$a=\frac{140400-1200\sqrt{12649}}{2600}$$

Divide 140400-1200\sqrt{12649} by 2600.

$$a=-\frac{6\sqrt{12649}}{13}+54$$

The equation is now solved.

$$a=\frac{6\sqrt{12649}}{13}+54 a=-\frac{6\sqrt{12649}}{13}+54$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$1300aa=a\times 150000-\left(9600a+288000\right)$$

Multiply a and a to get a^{2}.

$$1300a^{2}=a\times 150000-\left(9600a+288000\right)$$

To find the opposite of 9600a+288000, find the opposite of each term.

$$1300a^{2}=a\times 150000-9600a-288000$$

Combine a\times 150000 and -9600a to get 140400a.

$$1300a^{2}=140400a-288000$$

Subtract 140400a from both sides.

$$1300a^{2}-140400a=-288000$$

Divide both sides by 1300.

$$\frac{1300a^{2}-140400a}{1300}=-\frac{288000}{1300}$$

Dividing by 1300 undoes the multiplication by 1300.

$$a^{2}+\left(-\frac{140400}{1300}\right)a=-\frac{288000}{1300}$$

Divide -140400 by 1300.

$$a^{2}-108a=-\frac{288000}{1300}$$

Reduce the fraction \frac{-288000}{1300} to lowest terms by extracting and canceling out 100.

$$a^{2}-108a=-\frac{2880}{13}$$

Divide -108, the coefficient of the x term, by 2 to get -54. Then add the square of -54 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-108a+\left(-54\right)^{2}=-\frac{2880}{13}+\left(-54\right)^{2}$$

Square -54.

$$a^{2}-108a+2916=-\frac{2880}{13}+2916$$

Add -\frac{2880}{13} to 2916.

$$a^{2}-108a+2916=\frac{35028}{13}$$

Factor a^{2}-108a+2916. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-54\right)^{2}=\frac{35028}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-54\right)^{2}}=\sqrt{\frac{35028}{13}}$$

Simplify.

$$a-54=\frac{6\sqrt{12649}}{13} a-54=-\frac{6\sqrt{12649}}{13}$$

Add 54 to both sides of the equation.

$$a=\frac{6\sqrt{12649}}{13}+54 a=-\frac{6\sqrt{12649}}{13}+54$$