Solve 9000*(1+x/2%)^2=9548.1

$$9000 \times ( 1 + \frac { x } { 2 } \% ) ^ { 2 } = 9548.1$$

Solve for x

$$x=6$$

$$x=-406$$

Steps Using Factoring

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548.1}{9000}$$

Expand \frac{9548.1}{9000} by multiplying both numerator and the denominator by 10.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{95481}{90000}$$

Reduce the fraction \frac{95481}{90000} to lowest terms by extracting and canceling out 9.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{10609}{10000}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{10609}{10000}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{10609}{10000}$$

Subtract \frac{10609}{10000} from both sides.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}-\frac{10609}{10000}=0$$

Subtract \frac{10609}{10000} from 1 to get -\frac{609}{10000}.

$$-\frac{609}{10000}+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=0$$

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

$$-609+200\times \frac{x}{2}+\frac{x^{2}}{4}=0$$

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

$$-2436+400x+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+400x-2436=0$$

To solve the equation, factor x^{2}+400x-2436 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=400 ab=-2436$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2436.

$$-1,2436 -2,1218 -3,812 -4,609 -6,406 -7,348 -12,203 -14,174 -21,116 -28,87 -29,84 -42,58$$

Calculate the sum for each pair.

$$-1+2436=2435 -2+1218=1216 -3+812=809 -4+609=605 -6+406=400 -7+348=341 -12+203=191 -14+174=160 -21+116=95 -28+87=59 -29+84=55 -42+58=16$$

The solution is the pair that gives sum 400.

$$a=-6 b=406$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-6\right)\left(x+406\right)$$

To find equation solutions, solve x-6=0 and x+406=0.

$$x=6 x=-406$$

Steps Using Factoring By Grouping

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548.1}{9000}$$

Expand \frac{9548.1}{9000} by multiplying both numerator and the denominator by 10.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{95481}{90000}$$

Reduce the fraction \frac{95481}{90000} to lowest terms by extracting and canceling out 9.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{10609}{10000}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{10609}{10000}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{10609}{10000}$$

Subtract \frac{10609}{10000} from both sides.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}-\frac{10609}{10000}=0$$

Subtract \frac{10609}{10000} from 1 to get -\frac{609}{10000}.

$$-\frac{609}{10000}+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=0$$

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

$$-609+200\times \frac{x}{2}+\frac{x^{2}}{4}=0$$

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

$$-2436+400x+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+400x-2436=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-2436. To find a and b, set up a system to be solved.

$$a+b=400 ab=1\left(-2436\right)=-2436$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2436.

$$-1,2436 -2,1218 -3,812 -4,609 -6,406 -7,348 -12,203 -14,174 -21,116 -28,87 -29,84 -42,58$$

Calculate the sum for each pair.

$$-1+2436=2435 -2+1218=1216 -3+812=809 -4+609=605 -6+406=400 -7+348=341 -12+203=191 -14+174=160 -21+116=95 -28+87=59 -29+84=55 -42+58=16$$

The solution is the pair that gives sum 400.

$$a=-6 b=406$$

Rewrite x^{2}+400x-2436 as \left(x^{2}-6x\right)+\left(406x-2436\right).

$$\left(x^{2}-6x\right)+\left(406x-2436\right)$$

Factor out x in the first and 406 in the second group.

$$x\left(x-6\right)+406\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+406\right)$$

To find equation solutions, solve x-6=0 and x+406=0.

$$x=6 x=-406$$

Steps Using the Quadratic Formula

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548.1}{9000}$$

Expand \frac{9548.1}{9000} by multiplying both numerator and the denominator by 10.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{95481}{90000}$$

Reduce the fraction \frac{95481}{90000} to lowest terms by extracting and canceling out 9.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{10609}{10000}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{10609}{10000}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{10609}{10000}$$

Subtract \frac{10609}{10000} from both sides.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}-\frac{10609}{10000}=0$$

Subtract \frac{10609}{10000} from 1 to get -\frac{609}{10000}.

$$-\frac{609}{10000}+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=0$$

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

$$-609+200\times \frac{x}{2}+\frac{x^{2}}{4}=0$$

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

$$-2436+400x+x^{2}=0$$

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.

$$x^{2}+400x-2436=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 400 for b, and -2436 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\left(-2436\right)}}{2}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\left(-2436\right)}}{2}$$

Multiply -4 times -2436.

$$x=\frac{-400±\sqrt{160000+9744}}{2}$$

Add 160000 to 9744.

$$x=\frac{-400±\sqrt{169744}}{2}$$

Take the square root of 169744.

$$x=\frac{-400±412}{2}$$

Now solve the equation x=\frac{-400±412}{2} when ± is plus. Add -400 to 412.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{-400±412}{2} when ± is minus. Subtract 412 from -400.

$$x=-\frac{812}{2}$$

Divide -812 by 2.

$$x=-406$$

The equation is now solved.

$$x=6 x=-406$$

Steps for Completing the Square

Divide both sides by 9000.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{9548.1}{9000}$$

Expand \frac{9548.1}{9000} by multiplying both numerator and the denominator by 10.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{95481}{90000}$$

Reduce the fraction \frac{95481}{90000} to lowest terms by extracting and canceling out 9.

$$\left(1+\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\left(\frac{\frac{x}{2}}{100}\right)^{2}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\left(\frac{x}{2}\right)^{2}}{100^{2}}=\frac{10609}{10000}$$

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

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{100^{2}}=\frac{10609}{10000}$$

Calculate 100 to the power of 2 and get 10000.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{2^{2}}}{10000}=\frac{10609}{10000}$$

Calculate 2 to the power of 2 and get 4.

$$1+2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{10609}{10000}$$

Subtract 1 from both sides.

$$2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{10609}{10000}-1$$

Subtract 1 from \frac{10609}{10000} to get \frac{609}{10000}.

$$2\times \frac{\frac{x}{2}}{100}+\frac{\frac{x^{2}}{4}}{10000}=\frac{609}{10000}$$

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

$$200\times \frac{x}{2}+\frac{x^{2}}{4}=609$$

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

$$400x+x^{2}=2436$$

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.

$$x^{2}+400x=2436$$

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.

$$x^{2}+400x+200^{2}=2436+200^{2}$$

Square 200.

$$x^{2}+400x+40000=2436+40000$$

Add 2436 to 40000.

$$x^{2}+400x+40000=42436$$

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}.

$$\left(x+200\right)^{2}=42436$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+200\right)^{2}}=\sqrt{42436}$$

Simplify.

$$x+200=206 x+200=-206$$

Subtract 200 from both sides of the equation.

$$x=6 x=-406$$