Solve 100=20t+1/2*9.8t^2

$$100 = 20 t + \frac { 1 } { 2 } \times 9.8 t ^ { 2 }$$

Solve for t

$$t = \frac{10 \sqrt{590} - 100}{49} \approx 2.916309307$$

$$t=\frac{-10\sqrt{590}-100}{49}\approx -6.99794196$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 9.8 to get \frac{49}{10}.

$$100=20t+\frac{49}{10}t^{2}$$

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

$$20t+\frac{49}{10}t^{2}=100$$

Subtract 100 from both sides.

$$20t+\frac{49}{10}t^{2}-100=0$$

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.

$$\frac{49}{10}t^{2}+20t-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{49}{10} for a, 20 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\times \frac{49}{10}\left(-100\right)}}{2\times \frac{49}{10}}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\times \frac{49}{10}\left(-100\right)}}{2\times \frac{49}{10}}$$

Multiply -4 times \frac{49}{10}.

$$t=\frac{-20±\sqrt{400-\frac{98}{5}\left(-100\right)}}{2\times \frac{49}{10}}$$

Multiply -\frac{98}{5} times -100.

$$t=\frac{-20±\sqrt{400+1960}}{2\times \frac{49}{10}}$$

Add 400 to 1960.

$$t=\frac{-20±\sqrt{2360}}{2\times \frac{49}{10}}$$

Take the square root of 2360.

$$t=\frac{-20±2\sqrt{590}}{2\times \frac{49}{10}}$$

Multiply 2 times \frac{49}{10}.

$$t=\frac{-20±2\sqrt{590}}{\frac{49}{5}}$$

Now solve the equation t=\frac{-20±2\sqrt{590}}{\frac{49}{5}} when ± is plus. Add -20 to 2\sqrt{590}.

$$t=\frac{2\sqrt{590}-20}{\frac{49}{5}}$$

Divide -20+2\sqrt{590} by \frac{49}{5} by multiplying -20+2\sqrt{590} by the reciprocal of \frac{49}{5}.

$$t=\frac{10\sqrt{590}-100}{49}$$

Now solve the equation t=\frac{-20±2\sqrt{590}}{\frac{49}{5}} when ± is minus. Subtract 2\sqrt{590} from -20.

$$t=\frac{-2\sqrt{590}-20}{\frac{49}{5}}$$

Divide -20-2\sqrt{590} by \frac{49}{5} by multiplying -20-2\sqrt{590} by the reciprocal of \frac{49}{5}.

$$t=\frac{-10\sqrt{590}-100}{49}$$

The equation is now solved.

$$t=\frac{10\sqrt{590}-100}{49} t=\frac{-10\sqrt{590}-100}{49}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 9.8 to get \frac{49}{10}.

$$100=20t+\frac{49}{10}t^{2}$$

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

$$20t+\frac{49}{10}t^{2}=100$$

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.

$$\frac{49}{10}t^{2}+20t=100$$

Divide both sides of the equation by \frac{49}{10}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{49}{10}t^{2}+20t}{\frac{49}{10}}=\frac{100}{\frac{49}{10}}$$

Dividing by \frac{49}{10} undoes the multiplication by \frac{49}{10}.

$$t^{2}+\frac{20}{\frac{49}{10}}t=\frac{100}{\frac{49}{10}}$$

Divide 20 by \frac{49}{10} by multiplying 20 by the reciprocal of \frac{49}{10}.

$$t^{2}+\frac{200}{49}t=\frac{100}{\frac{49}{10}}$$

Divide 100 by \frac{49}{10} by multiplying 100 by the reciprocal of \frac{49}{10}.

$$t^{2}+\frac{200}{49}t=\frac{1000}{49}$$

Divide \frac{200}{49}, the coefficient of the x term, by 2 to get \frac{100}{49}. Then add the square of \frac{100}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{200}{49}t+\left(\frac{100}{49}\right)^{2}=\frac{1000}{49}+\left(\frac{100}{49}\right)^{2}$$

Square \frac{100}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{200}{49}t+\frac{10000}{2401}=\frac{1000}{49}+\frac{10000}{2401}$$

Add \frac{1000}{49} to \frac{10000}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{200}{49}t+\frac{10000}{2401}=\frac{59000}{2401}$$

Factor t^{2}+\frac{200}{49}t+\frac{10000}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{100}{49}\right)^{2}=\frac{59000}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{100}{49}\right)^{2}}=\sqrt{\frac{59000}{2401}}$$

Simplify.

$$t+\frac{100}{49}=\frac{10\sqrt{590}}{49} t+\frac{100}{49}=-\frac{10\sqrt{590}}{49}$$

Subtract \frac{100}{49} from both sides of the equation.

$$t=\frac{10\sqrt{590}-100}{49} t=\frac{-10\sqrt{590}-100}{49}$$