Solve 30000=12t^2+800t+40000

$$30000 = 12 { t }^{ 2 } +800t+40000$$

Solve for t

$$t = -\frac{50}{3} = -16\frac{2}{3} \approx -16.666666667$$

$$t=-50$$

Steps Using the Quadratic Formula

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

$$12t^{2}+800t+40000=30000$$

Subtract 30000 from both sides.

$$12t^{2}+800t+40000-30000=0$$

Subtract 30000 from 40000 to get 10000.

$$12t^{2}+800t+10000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 800 for b, and 10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-800±\sqrt{800^{2}-4\times 12\times 10000}}{2\times 12}$$

Square 800.

$$t=\frac{-800±\sqrt{640000-4\times 12\times 10000}}{2\times 12}$$

Multiply -4 times 12.

$$t=\frac{-800±\sqrt{640000-48\times 10000}}{2\times 12}$$

Multiply -48 times 10000.

$$t=\frac{-800±\sqrt{640000-480000}}{2\times 12}$$

Add 640000 to -480000.

$$t=\frac{-800±\sqrt{160000}}{2\times 12}$$

Take the square root of 160000.

$$t=\frac{-800±400}{2\times 12}$$

Multiply 2 times 12.

$$t=\frac{-800±400}{24}$$

Now solve the equation t=\frac{-800±400}{24} when ± is plus. Add -800 to 400.

$$t=-\frac{400}{24}$$

Reduce the fraction \frac{-400}{24} to lowest terms by extracting and canceling out 8.

$$t=-\frac{50}{3}$$

Now solve the equation t=\frac{-800±400}{24} when ± is minus. Subtract 400 from -800.

$$t=-\frac{1200}{24}$$

Divide -1200 by 24.

$$t=-50$$

The equation is now solved.

$$t=-\frac{50}{3} t=-50$$

Steps for Completing the Square

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

$$12t^{2}+800t+40000=30000$$

Subtract 40000 from both sides.

$$12t^{2}+800t=30000-40000$$

Subtract 40000 from 30000 to get -10000.

$$12t^{2}+800t=-10000$$

Divide both sides by 12.

$$\frac{12t^{2}+800t}{12}=-\frac{10000}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$t^{2}+\frac{800}{12}t=-\frac{10000}{12}$$

Reduce the fraction \frac{800}{12} to lowest terms by extracting and canceling out 4.

$$t^{2}+\frac{200}{3}t=-\frac{10000}{12}$$

Reduce the fraction \frac{-10000}{12} to lowest terms by extracting and canceling out 4.

$$t^{2}+\frac{200}{3}t=-\frac{2500}{3}$$

Divide \frac{200}{3}, the coefficient of the x term, by 2 to get \frac{100}{3}. Then add the square of \frac{100}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{200}{3}t+\left(\frac{100}{3}\right)^{2}=-\frac{2500}{3}+\left(\frac{100}{3}\right)^{2}$$

Square \frac{100}{3} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{200}{3}t+\frac{10000}{9}=-\frac{2500}{3}+\frac{10000}{9}$$

Add -\frac{2500}{3} to \frac{10000}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{200}{3}t+\frac{10000}{9}=\frac{2500}{9}$$

Factor t^{2}+\frac{200}{3}t+\frac{10000}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{100}{3}\right)^{2}=\frac{2500}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{100}{3}\right)^{2}}=\sqrt{\frac{2500}{9}}$$

Simplify.

$$t+\frac{100}{3}=\frac{50}{3} t+\frac{100}{3}=-\frac{50}{3}$$

Subtract \frac{100}{3} from both sides of the equation.

$$t=-\frac{50}{3} t=-50$$