Solve 120=3/10^6(x)(24000-x)

$$120 = \frac { 3 } { 10 ^ { 6 } } ( x ) ( 24000 - x )$$

Solve for x

$$x=2000\sqrt{26}+12000\approx 22198.039027186$$

$$x=12000-2000\sqrt{26}\approx 1801.960972814$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 6 and get 1000000.

$$120=\frac{3}{1000000}x\left(24000-x\right)$$

Use the distributive property to multiply \frac{3}{1000000}x by 24000-x.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x\left(-1\right)x$$

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

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x^{2}\left(-1\right)$$

Express \frac{3}{1000000}\times 24000 as a single fraction.

$$120=\frac{3\times 24000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Multiply 3 and 24000 to get 72000.

$$120=\frac{72000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Reduce the fraction \frac{72000}{1000000} to lowest terms by extracting and canceling out 8000.

$$120=\frac{9}{125}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Multiply \frac{3}{1000000} and -1 to get -\frac{3}{1000000}.

$$120=\frac{9}{125}x-\frac{3}{1000000}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}=120$$

Subtract 120 from both sides.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}-120=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-\frac{3}{1000000}x^{2}+\frac{9}{125}x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{1000000} for a, \frac{9}{125} for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{125}±\sqrt{\left(\frac{9}{125}\right)^{2}-4\left(-\frac{3}{1000000}\right)\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Square \frac{9}{125} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}-4\left(-\frac{3}{1000000}\right)\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Multiply -4 times -\frac{3}{1000000}.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}+\frac{3}{250000}\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Multiply \frac{3}{250000} times -120.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}-\frac{9}{6250}}}{2\left(-\frac{3}{1000000}\right)}$$

Add \frac{81}{15625} to -\frac{9}{6250} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{117}{31250}}}{2\left(-\frac{3}{1000000}\right)}$$

Take the square root of \frac{117}{31250}.

$$x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{2\left(-\frac{3}{1000000}\right)}$$

Multiply 2 times -\frac{3}{1000000}.

$$x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}}$$

Now solve the equation x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}} when ± is plus. Add -\frac{9}{125} to \frac{3\sqrt{26}}{250}.

$$x=\frac{\frac{3\sqrt{26}}{250}-\frac{9}{125}}{-\frac{3}{500000}}$$

Divide -\frac{9}{125}+\frac{3\sqrt{26}}{250} by -\frac{3}{500000} by multiplying -\frac{9}{125}+\frac{3\sqrt{26}}{250} by the reciprocal of -\frac{3}{500000}.

$$x=12000-2000\sqrt{26}$$

Now solve the equation x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}} when ± is minus. Subtract \frac{3\sqrt{26}}{250} from -\frac{9}{125}.

$$x=\frac{-\frac{3\sqrt{26}}{250}-\frac{9}{125}}{-\frac{3}{500000}}$$

Divide -\frac{9}{125}-\frac{3\sqrt{26}}{250} by -\frac{3}{500000} by multiplying -\frac{9}{125}-\frac{3\sqrt{26}}{250} by the reciprocal of -\frac{3}{500000}.

$$x=2000\sqrt{26}+12000$$

The equation is now solved.

$$x=12000-2000\sqrt{26} x=2000\sqrt{26}+12000$$

Steps for Completing the Square

Calculate 10 to the power of 6 and get 1000000.

$$120=\frac{3}{1000000}x\left(24000-x\right)$$

Use the distributive property to multiply \frac{3}{1000000}x by 24000-x.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x\left(-1\right)x$$

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

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x^{2}\left(-1\right)$$

Express \frac{3}{1000000}\times 24000 as a single fraction.

$$120=\frac{3\times 24000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Multiply 3 and 24000 to get 72000.

$$120=\frac{72000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Reduce the fraction \frac{72000}{1000000} to lowest terms by extracting and canceling out 8000.

$$120=\frac{9}{125}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Multiply \frac{3}{1000000} and -1 to get -\frac{3}{1000000}.

$$120=\frac{9}{125}x-\frac{3}{1000000}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}=120$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-\frac{3}{1000000}x^{2}+\frac{9}{125}x=120$$

Divide both sides of the equation by -\frac{3}{1000000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{3}{1000000}x^{2}+\frac{9}{125}x}{-\frac{3}{1000000}}=\frac{120}{-\frac{3}{1000000}}$$

Dividing by -\frac{3}{1000000} undoes the multiplication by -\frac{3}{1000000}.

$$x^{2}+\frac{\frac{9}{125}}{-\frac{3}{1000000}}x=\frac{120}{-\frac{3}{1000000}}$$

Divide \frac{9}{125} by -\frac{3}{1000000} by multiplying \frac{9}{125} by the reciprocal of -\frac{3}{1000000}.

$$x^{2}-24000x=\frac{120}{-\frac{3}{1000000}}$$

Divide 120 by -\frac{3}{1000000} by multiplying 120 by the reciprocal of -\frac{3}{1000000}.

$$x^{2}-24000x=-40000000$$

Divide -24000, the coefficient of the x term, by 2 to get -12000. Then add the square of -12000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24000x+\left(-12000\right)^{2}=-40000000+\left(-12000\right)^{2}$$

Square -12000.

$$x^{2}-24000x+144000000=-40000000+144000000$$

Add -40000000 to 144000000.

$$x^{2}-24000x+144000000=104000000$$

Factor x^{2}-24000x+144000000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12000\right)^{2}=104000000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12000\right)^{2}}=\sqrt{104000000}$$

Simplify.

$$x-12000=2000\sqrt{26} x-12000=-2000\sqrt{26}$$

Add 12000 to both sides of the equation.

$$x=2000\sqrt{26}+12000 x=12000-2000\sqrt{26}$$