Solve 2=6.45t-1/2*9.8*t^2

$$2 = 6.45 t - \frac { 1 } { 2 } \times 9.8 \times t ^ { 2 }$$

Solve for t

$$t=0.5$$

$$t=\frac{40}{49}\approx 0.816326531$$

Steps Using the Quadratic Formula

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

$$2=6.45t-\frac{49}{10}t^{2}$$

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

$$6.45t-\frac{49}{10}t^{2}=2$$

Subtract 2 from both sides.

$$6.45t-\frac{49}{10}t^{2}-2=0$$

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.

$$-\frac{49}{10}t^{2}+6.45t-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{49}{10} for a, 6.45 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-6.45±\sqrt{6.45^{2}-4\left(-\frac{49}{10}\right)\left(-2\right)}}{2\left(-\frac{49}{10}\right)}$$

Square 6.45 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-6.45±\sqrt{41.6025-4\left(-\frac{49}{10}\right)\left(-2\right)}}{2\left(-\frac{49}{10}\right)}$$

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

$$t=\frac{-6.45±\sqrt{41.6025+\frac{98}{5}\left(-2\right)}}{2\left(-\frac{49}{10}\right)}$$

Multiply \frac{98}{5} times -2.

$$t=\frac{-6.45±\sqrt{41.6025-\frac{196}{5}}}{2\left(-\frac{49}{10}\right)}$$

Add 41.6025 to -\frac{196}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-6.45±\sqrt{\frac{961}{400}}}{2\left(-\frac{49}{10}\right)}$$

Take the square root of \frac{961}{400}.

$$t=\frac{-6.45±\frac{31}{20}}{2\left(-\frac{49}{10}\right)}$$

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

$$t=\frac{-6.45±\frac{31}{20}}{-\frac{49}{5}}$$

Now solve the equation t=\frac{-6.45±\frac{31}{20}}{-\frac{49}{5}} when ± is plus. Add -6.45 to \frac{31}{20} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{\frac{49}{10}}{-\frac{49}{5}}$$

Divide -\frac{49}{10} by -\frac{49}{5} by multiplying -\frac{49}{10} by the reciprocal of -\frac{49}{5}.

$$t=\frac{1}{2}$$

Now solve the equation t=\frac{-6.45±\frac{31}{20}}{-\frac{49}{5}} when ± is minus. Subtract \frac{31}{20} from -6.45 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{8}{-\frac{49}{5}}$$

Divide -8 by -\frac{49}{5} by multiplying -8 by the reciprocal of -\frac{49}{5}.

$$t=\frac{40}{49}$$

The equation is now solved.

$$t=\frac{1}{2} t=\frac{40}{49}$$

Steps for Completing the Square

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

$$2=6.45t-\frac{49}{10}t^{2}$$

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

$$6.45t-\frac{49}{10}t^{2}=2$$

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.

$$-\frac{49}{10}t^{2}+6.45t=2$$

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

$$\frac{-\frac{49}{10}t^{2}+6.45t}{-\frac{49}{10}}=\frac{2}{-\frac{49}{10}}$$

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

$$t^{2}+\frac{6.45}{-\frac{49}{10}}t=\frac{2}{-\frac{49}{10}}$$

Divide 6.45 by -\frac{49}{10} by multiplying 6.45 by the reciprocal of -\frac{49}{10}.

$$t^{2}-\frac{129}{98}t=\frac{2}{-\frac{49}{10}}$$

Divide 2 by -\frac{49}{10} by multiplying 2 by the reciprocal of -\frac{49}{10}.

$$t^{2}-\frac{129}{98}t=-\frac{20}{49}$$

Divide -\frac{129}{98}, the coefficient of the x term, by 2 to get -\frac{129}{196}. Then add the square of -\frac{129}{196} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{129}{98}t+\left(-\frac{129}{196}\right)^{2}=-\frac{20}{49}+\left(-\frac{129}{196}\right)^{2}$$

Square -\frac{129}{196} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{129}{98}t+\frac{16641}{38416}=-\frac{20}{49}+\frac{16641}{38416}$$

Add -\frac{20}{49} to \frac{16641}{38416} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{129}{98}t+\frac{16641}{38416}=\frac{961}{38416}$$

Factor t^{2}-\frac{129}{98}t+\frac{16641}{38416}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{129}{196}\right)^{2}=\frac{961}{38416}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{129}{196}\right)^{2}}=\sqrt{\frac{961}{38416}}$$

Simplify.

$$t-\frac{129}{196}=\frac{31}{196} t-\frac{129}{196}=-\frac{31}{196}$$

Add \frac{129}{196} to both sides of the equation.

$$t=\frac{40}{49} t=\frac{1}{2}$$