Solve 0=r^2+19r-2140

$$0 = r ^ { 2 } + 19 r - 2140$$

Solve for r

$$r = \frac{\sqrt{8921} - 19}{2} \approx 37.72552276$$

$$r=\frac{-\sqrt{8921}-19}{2}\approx -56.72552276$$

Steps Using the Quadratic Formula

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

$$r^{2}+19r-2140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and -2140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-19±\sqrt{19^{2}-4\left(-2140\right)}}{2}$$

Square 19.

$$r=\frac{-19±\sqrt{361-4\left(-2140\right)}}{2}$$

Multiply -4 times -2140.

$$r=\frac{-19±\sqrt{361+8560}}{2}$$

Add 361 to 8560.

$$r=\frac{-19±\sqrt{8921}}{2}$$

Now solve the equation r=\frac{-19±\sqrt{8921}}{2} when ± is plus. Add -19 to \sqrt{8921}.

$$r=\frac{\sqrt{8921}-19}{2}$$

Now solve the equation r=\frac{-19±\sqrt{8921}}{2} when ± is minus. Subtract \sqrt{8921} from -19.

$$r=\frac{-\sqrt{8921}-19}{2}$$

The equation is now solved.

$$r=\frac{\sqrt{8921}-19}{2} r=\frac{-\sqrt{8921}-19}{2}$$

Steps for Completing the Square

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

$$r^{2}+19r-2140=0$$

Add 2140 to both sides. Anything plus zero gives itself.

$$r^{2}+19r=2140$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+19r+\left(\frac{19}{2}\right)^{2}=2140+\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+19r+\frac{361}{4}=2140+\frac{361}{4}$$

Add 2140 to \frac{361}{4}.

$$r^{2}+19r+\frac{361}{4}=\frac{8921}{4}$$

Factor r^{2}+19r+\frac{361}{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}.

$$\left(r+\frac{19}{2}\right)^{2}=\frac{8921}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{19}{2}\right)^{2}}=\sqrt{\frac{8921}{4}}$$

Simplify.

$$r+\frac{19}{2}=\frac{\sqrt{8921}}{2} r+\frac{19}{2}=-\frac{\sqrt{8921}}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$r=\frac{\sqrt{8921}-19}{2} r=\frac{-\sqrt{8921}-19}{2}$$