Solve 17,494=12,696(1+r)^2

$$17,494 = 12,696 ( 1 + r ) ^ { 2 }$$

Solve for r

$$r=\frac{\sqrt{26241}}{138}-1\approx 0.173845945$$

$$r=-\frac{\sqrt{26241}}{138}-1\approx -2.173845945$$

Steps Using the Quadratic Formula

Divide both sides by 12696.

$$\frac{17494}{12696}=\left(1+r\right)^{2}$$

Reduce the fraction \frac{17494}{12696} to lowest terms by extracting and canceling out 2.

$$\frac{8747}{6348}=\left(1+r\right)^{2}$$

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

$$\frac{8747}{6348}=1+2r+r^{2}$$

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

$$1+2r+r^{2}=\frac{8747}{6348}$$

Subtract \frac{8747}{6348} from both sides.

$$1+2r+r^{2}-\frac{8747}{6348}=0$$

Subtract \frac{8747}{6348} from 1 to get -\frac{2399}{6348}.

$$-\frac{2399}{6348}+2r+r^{2}=0$$

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.

$$r^{2}+2r-\frac{2399}{6348}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -\frac{2399}{6348} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-2±\sqrt{2^{2}-4\left(-\frac{2399}{6348}\right)}}{2}$$

Square 2.

$$r=\frac{-2±\sqrt{4-4\left(-\frac{2399}{6348}\right)}}{2}$$

Multiply -4 times -\frac{2399}{6348}.

$$r=\frac{-2±\sqrt{4+\frac{2399}{1587}}}{2}$$

Add 4 to \frac{2399}{1587}.

$$r=\frac{-2±\sqrt{\frac{8747}{1587}}}{2}$$

Take the square root of \frac{8747}{1587}.

$$r=\frac{-2±\frac{\sqrt{26241}}{69}}{2}$$

Now solve the equation r=\frac{-2±\frac{\sqrt{26241}}{69}}{2} when ± is plus. Add -2 to \frac{\sqrt{26241}}{69}.

$$r=\frac{\frac{\sqrt{26241}}{69}-2}{2}$$

Divide -2+\frac{\sqrt{26241}}{69} by 2.

$$r=\frac{\sqrt{26241}}{138}-1$$

Now solve the equation r=\frac{-2±\frac{\sqrt{26241}}{69}}{2} when ± is minus. Subtract \frac{\sqrt{26241}}{69} from -2.

$$r=\frac{-\frac{\sqrt{26241}}{69}-2}{2}$$

Divide -2-\frac{\sqrt{26241}}{69} by 2.

$$r=-\frac{\sqrt{26241}}{138}-1$$

The equation is now solved.

$$r=\frac{\sqrt{26241}}{138}-1 r=-\frac{\sqrt{26241}}{138}-1$$

Steps for Completing the Square

Divide both sides by 12696.

$$\frac{17494}{12696}=\left(1+r\right)^{2}$$

Reduce the fraction \frac{17494}{12696} to lowest terms by extracting and canceling out 2.

$$\frac{8747}{6348}=\left(1+r\right)^{2}$$

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

$$\frac{8747}{6348}=1+2r+r^{2}$$

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

$$1+2r+r^{2}=\frac{8747}{6348}$$

Subtract 1 from both sides.

$$2r+r^{2}=\frac{8747}{6348}-1$$

Subtract 1 from \frac{8747}{6348} to get \frac{2399}{6348}.

$$2r+r^{2}=\frac{2399}{6348}$$

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.

$$r^{2}+2r=\frac{2399}{6348}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+2r+1^{2}=\frac{2399}{6348}+1^{2}$$

Square 1.

$$r^{2}+2r+1=\frac{2399}{6348}+1$$

Add \frac{2399}{6348} to 1.

$$r^{2}+2r+1=\frac{8747}{6348}$$

Factor r^{2}+2r+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+1\right)^{2}=\frac{8747}{6348}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+1\right)^{2}}=\sqrt{\frac{8747}{6348}}$$

Simplify.

$$r+1=\frac{\sqrt{26241}}{138} r+1=-\frac{\sqrt{26241}}{138}$$

Subtract 1 from both sides of the equation.

$$r=\frac{\sqrt{26241}}{138}-1 r=-\frac{\sqrt{26241}}{138}-1$$