Solve 15000+600=15000{left(1+r/2right)}^2*1

$$15000+600 = 15000 { \left(1+ \frac{ r }{ 2 } \right) }^{ 2 \times 1 }$$

Solve for r

$$r=\frac{2\sqrt{26}}{5}-2\approx 0.039607805$$

$$r=-\frac{2\sqrt{26}}{5}-2\approx -4.039607805$$

Steps Using the Quadratic Formula

Add 15000 and 600 to get 15600.

$$15600=15000\left(1+\frac{r}{2}\right)^{2\times 1}$$

Multiply 2 and 1 to get 2.

$$15600=15000\left(1+\frac{r}{2}\right)^{2}$$

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

$$15600=15000\left(1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}\right)$$

Express 2\times \frac{r}{2} as a single fraction.

$$15600=15000\left(1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}\right)$$

Cancel out 2 and 2.

$$15600=15000\left(1+r+\left(\frac{r}{2}\right)^{2}\right)$$

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

$$15600=15000\left(1+r+\frac{r^{2}}{2^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1+r times \frac{2^{2}}{2^{2}}.

$$15600=15000\left(\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}\right)$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$15600=15000\times \frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$15600=15000\times \frac{4+4r+r^{2}}{2^{2}}$$

Express 15000\times \frac{4+4r+r^{2}}{2^{2}} as a single fraction.

$$15600=\frac{15000\left(4+4r+r^{2}\right)}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$15600=\frac{15000\left(4+4r+r^{2}\right)}{4}$$

Divide 15000\left(4+4r+r^{2}\right) by 4 to get 3750\left(4+4r+r^{2}\right).

$$15600=3750\left(4+4r+r^{2}\right)$$

Use the distributive property to multiply 3750 by 4+4r+r^{2}.

$$15600=15000+15000r+3750r^{2}$$

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

$$15000+15000r+3750r^{2}=15600$$

Subtract 15600 from both sides.

$$15000+15000r+3750r^{2}-15600=0$$

Subtract 15600 from 15000 to get -600.

$$-600+15000r+3750r^{2}=0$$

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.

$$3750r^{2}+15000r-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3750 for a, 15000 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-15000±\sqrt{15000^{2}-4\times 3750\left(-600\right)}}{2\times 3750}$$

Square 15000.

$$r=\frac{-15000±\sqrt{225000000-4\times 3750\left(-600\right)}}{2\times 3750}$$

Multiply -4 times 3750.

$$r=\frac{-15000±\sqrt{225000000-15000\left(-600\right)}}{2\times 3750}$$

Multiply -15000 times -600.

$$r=\frac{-15000±\sqrt{225000000+9000000}}{2\times 3750}$$

Add 225000000 to 9000000.

$$r=\frac{-15000±\sqrt{234000000}}{2\times 3750}$$

Take the square root of 234000000.

$$r=\frac{-15000±3000\sqrt{26}}{2\times 3750}$$

Multiply 2 times 3750.

$$r=\frac{-15000±3000\sqrt{26}}{7500}$$

Now solve the equation r=\frac{-15000±3000\sqrt{26}}{7500} when ± is plus. Add -15000 to 3000\sqrt{26}.

$$r=\frac{3000\sqrt{26}-15000}{7500}$$

Divide -15000+3000\sqrt{26} by 7500.

$$r=\frac{2\sqrt{26}}{5}-2$$

Now solve the equation r=\frac{-15000±3000\sqrt{26}}{7500} when ± is minus. Subtract 3000\sqrt{26} from -15000.

$$r=\frac{-3000\sqrt{26}-15000}{7500}$$

Divide -15000-3000\sqrt{26} by 7500.

$$r=-\frac{2\sqrt{26}}{5}-2$$

The equation is now solved.

$$r=\frac{2\sqrt{26}}{5}-2 r=-\frac{2\sqrt{26}}{5}-2$$

Steps for Completing the Square

Add 15000 and 600 to get 15600.

$$15600=15000\left(1+\frac{r}{2}\right)^{2\times 1}$$

Multiply 2 and 1 to get 2.

$$15600=15000\left(1+\frac{r}{2}\right)^{2}$$

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

$$15600=15000\left(1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}\right)$$

Express 2\times \frac{r}{2} as a single fraction.

$$15600=15000\left(1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}\right)$$

Cancel out 2 and 2.

$$15600=15000\left(1+r+\left(\frac{r}{2}\right)^{2}\right)$$

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

$$15600=15000\left(1+r+\frac{r^{2}}{2^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1+r times \frac{2^{2}}{2^{2}}.

$$15600=15000\left(\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}\right)$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$15600=15000\times \frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$15600=15000\times \frac{4+4r+r^{2}}{2^{2}}$$

Express 15000\times \frac{4+4r+r^{2}}{2^{2}} as a single fraction.

$$15600=\frac{15000\left(4+4r+r^{2}\right)}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$15600=\frac{15000\left(4+4r+r^{2}\right)}{4}$$

Divide 15000\left(4+4r+r^{2}\right) by 4 to get 3750\left(4+4r+r^{2}\right).

$$15600=3750\left(4+4r+r^{2}\right)$$

Use the distributive property to multiply 3750 by 4+4r+r^{2}.

$$15600=15000+15000r+3750r^{2}$$

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

$$15000+15000r+3750r^{2}=15600$$

Subtract 15000 from both sides.

$$15000r+3750r^{2}=15600-15000$$

Subtract 15000 from 15600 to get 600.

$$15000r+3750r^{2}=600$$

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.

$$3750r^{2}+15000r=600$$

Divide both sides by 3750.

$$\frac{3750r^{2}+15000r}{3750}=\frac{600}{3750}$$

Dividing by 3750 undoes the multiplication by 3750.

$$r^{2}+\frac{15000}{3750}r=\frac{600}{3750}$$

Divide 15000 by 3750.

$$r^{2}+4r=\frac{600}{3750}$$

Reduce the fraction \frac{600}{3750} to lowest terms by extracting and canceling out 150.

$$r^{2}+4r=\frac{4}{25}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+4r+2^{2}=\frac{4}{25}+2^{2}$$

Square 2.

$$r^{2}+4r+4=\frac{4}{25}+4$$

Add \frac{4}{25} to 4.

$$r^{2}+4r+4=\frac{104}{25}$$

Factor r^{2}+4r+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+2\right)^{2}=\frac{104}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+2\right)^{2}}=\sqrt{\frac{104}{25}}$$

Simplify.

$$r+2=\frac{2\sqrt{26}}{5} r+2=-\frac{2\sqrt{26}}{5}$$

Subtract 2 from both sides of the equation.

$$r=\frac{2\sqrt{26}}{5}-2 r=-\frac{2\sqrt{26}}{5}-2$$