Solve 9000*(1+x/2%)^2=9548

$$9000 \times ( 1 + \frac { x } { 2 } \% ) ^ { 2 } = 9548$$

Solve for x

$$x=\frac{4\sqrt{23870}}{3}-200\approx 5.998921249$$

$$x=-\frac{4\sqrt{23870}}{3}-200\approx -405.998921249$$

Steps Using the Quadratic Formula

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548}{9000}$$

Reduce the fraction \frac{9548}{9000} to lowest terms by extracting and canceling out 4.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{2387}{2250}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\frac{\frac{x}{2}}{100}\right)^{2}.

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{2387}{2250}$$

To raise \frac{\frac{x}{2}}{100} to a power, raise both numerator and denominator to the power and then divide.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{2387}{2250}$$

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{2387}{2250}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{2387}{2250}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{2387}{2250}$$

Subtract \frac{2387}{2250} from both sides.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}-\frac{2387}{2250}=0$$

Subtract \frac{2387}{2250} from 1 to get -\frac{137}{2250}.

$$-\frac{137}{2250}+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=0$$

Multiply both sides of the equation by 90000, the least common multiple of 2250,100,10000.

$$-5480+1800\times \frac{x}{2}+9\times \frac{x^{2}}{4}=0$$

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$-21920+3600x+9x^{2}=0$$

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.

$$9x^{2}+3600x-21920=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 3600 for b, and -21920 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3600±\sqrt{3600^{2}-4\times 9\left(-21920\right)}}{2\times 9}$$

Square 3600.

$$x=\frac{-3600±\sqrt{12960000-4\times 9\left(-21920\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-3600±\sqrt{12960000-36\left(-21920\right)}}{2\times 9}$$

Multiply -36 times -21920.

$$x=\frac{-3600±\sqrt{12960000+789120}}{2\times 9}$$

Add 12960000 to 789120.

$$x=\frac{-3600±\sqrt{13749120}}{2\times 9}$$

Take the square root of 13749120.

$$x=\frac{-3600±24\sqrt{23870}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-3600±24\sqrt{23870}}{18}$$

Now solve the equation x=\frac{-3600±24\sqrt{23870}}{18} when ± is plus. Add -3600 to 24\sqrt{23870}.

$$x=\frac{24\sqrt{23870}-3600}{18}$$

Divide -3600+24\sqrt{23870} by 18.

$$x=\frac{4\sqrt{23870}}{3}-200$$

Now solve the equation x=\frac{-3600±24\sqrt{23870}}{18} when ± is minus. Subtract 24\sqrt{23870} from -3600.

$$x=\frac{-24\sqrt{23870}-3600}{18}$$

Divide -3600-24\sqrt{23870} by 18.

$$x=-\frac{4\sqrt{23870}}{3}-200$$

The equation is now solved.

$$x=\frac{4\sqrt{23870}}{3}-200 x=-\frac{4\sqrt{23870}}{3}-200$$

Steps for Completing the Square

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548}{9000}$$

Reduce the fraction \frac{9548}{9000} to lowest terms by extracting and canceling out 4.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{2387}{2250}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\frac{\frac{x}{2}}{100}\right)^{2}.

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{2387}{2250}$$

To raise \frac{\frac{x}{2}}{100} to a power, raise both numerator and denominator to the power and then divide.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{2387}{2250}$$

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{2387}{2250}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{2387}{2250}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{2387}{2250}$$

Subtract 1 from both sides.

$$2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{2387}{2250}-1$$

Subtract 1 from \frac{2387}{2250} to get \frac{137}{2250}.

$$2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{137}{2250}$$

Multiply both sides of the equation by 90000, the least common multiple of 100,10000,2250.

$$1800\times \frac{x}{2}+9\times \frac{x^{2}}{4}=5480$$

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$3600x+9x^{2}=21920$$

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.

$$9x^{2}+3600x=21920$$

Divide both sides by 9.

$$\frac{9x^{2}+3600x}{9}=\frac{21920}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{3600}{9}x=\frac{21920}{9}$$

Divide 3600 by 9.

$$x^{2}+400x=\frac{21920}{9}$$

Divide 400, the coefficient of the x term, by 2 to get 200. Then add the square of 200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+400x+200^{2}=\frac{21920}{9}+200^{2}$$

Square 200.

$$x^{2}+400x+40000=\frac{21920}{9}+40000$$

Add \frac{21920}{9} to 40000.

$$x^{2}+400x+40000=\frac{381920}{9}$$

Factor x^{2}+400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+200\right)^{2}=\frac{381920}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+200\right)^{2}}=\sqrt{\frac{381920}{9}}$$

Simplify.

$$x+200=\frac{4\sqrt{23870}}{3} x+200=-\frac{4\sqrt{23870}}{3}$$

Subtract 200 from both sides of the equation.

$$x=\frac{4\sqrt{23870}}{3}-200 x=-\frac{4\sqrt{23870}}{3}-200$$